user15543: I recently played a game as Black. My opponent and I transposed into, according to Lichess, the Russian Game, Damiano Variation, when White played 4. Qe2. I wasn’t sure how to reply, and I ended up losing my queen and soon resigned.
[FEN ""]

1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nxe4 4. Qe2 Nf6 5. Nc6+ Be7 6. Nxd8 Kxd8

As Black, how should one respond to 4. Qe2 by White in the Russian Game, Damiano Variation?

user19027: In Damiano variant you respond 4. ... Qe7 to 4. Qe2. Yes, white takes the knight faster than you, but then you simply make d6 move, and if white knight retreats, white loses Queen. So to black's 5. ... d6 white have to respond with 6. d4, then: you take knight with the pawn and white makes a move dxe5. After those there's a bunch of possibilities that are better to be all checked on lichess.org via its analysis function.
Not to create any complications, you may not beat the e4 pawn after white knight grabs e5. Instead 3. ... d6 is a much better move, in my opinion, making white's knight retreat immediately, after which you can grab 4. ... Nxe4 without a thought after 4. Nf3, for example. And here to 5. Qe2 you simply respond 5. ... Qe7 and thus are able to retreat your knight next move without having a hidden check.

user3594: This part of the answer pertains to the first version of the post, where (due to a typo in setting up the diagram in OP) the move 4...f6 was played:
It started to go wrong for you from the move 4...f6, which just walks right into Qh5+ completely winning for white:
 [title "why 4...f6 is a mistake"]
 [fen "rnbqkb1r/pppp2pp/5p2/4N3/4n3/8/PPPPQPPP/RNB1KB1R w KQkq - 0 5"]

 5.Qh5+ g6 6.Nxg6 hxg6 7.Qxg6+ Ke7 8.Qxe4+ Kf7 9.Bc4+ d5 {forced to sac the pawn so Qg4 can be prevented} (9...Kg7 10.Qg4+ Kh6 11.d4+ Kh7 12.Bd3+ f5 13.Bxf5#) 10.Bxd5+ Kg7 11.Bxb7 Bxb7 12.Qxb7 Nd7 13.O-O {and white is completely winning, both on position and material count.} 

General advice: Particularly against 1.e4 systems, you almost always should try to avoid moving your f pawn early on, as white's queen and light square bishop are activated early on and therefore kingside light square weaknesses can be easily exploited. Typical to the Russian/Petroff move order, is rather d6 instead of f6 in order to kick the knight back. Though 4...d6 happens to also have been a poor choice in your game, as in that specific variation 4...Qe7 is really the only reasonable move that maintains the balance of the position. So let's look at that next:

The remaining part of the answer from here tackles the latest and corrected move order provided by the OP, with 4.Qe2 Qe7 5.Qxe4 and without f6:
 [title "4...Qe7: cover your king and set up a pin on e5 knight"]
 [fen "rnbqkb1r/pppp1ppp/8/4N3/4n3/8/PPPPQPPP/RNB1KB1R b KQkq - 1 4"]

 4...Qe7 5.Qxe4 d6 6.d4 dxe5 7.dxe5 Nc6 8.Bb5 Bd7 9.Nc3 O-O-O

And black has a much more reasonable play than they did in the aformentioned line.
Generally speaking, inherent to this opening's move order, moves such as Qe2 and Qe7 come quite naturally as the e file is opening up quickly and the discovery threats have to be parried, and these queen moves allow you to do so with tempo. Moreover, as you can see from the last line we looked at, trades are abound in this variation, and as black, you should not worry about dropping a pawn if you manage to compensate with development and piece activity (specially in short time control games). By sacrificing the d6 pawn on e5, we found a target pawn on e5 to coordinate against, which resolves your development plan (in contrast to white's play, hesitating as to whether defend the pawn or to regain initiative), and you have a slight initiative after the long castles. 

Addendum:
Upon further requests in the comments regarding the latest move order provided by the OP, here I provide additional explanations.
Important summary: In the Petroff, if you want to take white's pawn immediately after 3.Nxe5 then you should definitely expect 4.Qe2 as it is white's best move and black has really only one playable reply and that is 4...Qe7, as any other move either loses a piece or a pawn. For example the seemingly natural 4...d5 loses a pawn as follows:
 [title "4...d5 leads to a loss of pawn"]
 [fen "rnbqkb1r/pppp1ppp/8/4N3/4n3/8/PPPPQPPP/RNB1KB1R b KQkq - 1 4"]

 4...d5 5.d3 Qe7 6.dxe4 Qxe5 7.exd5

Therefore, it remains to cover all lines ensuing from 4...Qe7, but luckily, white's continuation is also quite limited if they want to play for an edge. The mainline that follows after 4...Qe7 is already covered in the 2nd diagram of my answer and the discussions thereafter. For completeness, it's worth pointing out that if in the said line (2nd diagram) white captures the pawn on e5 with the queen on move 7.Qxe5, then black has two choices:

Avoiding the trade of queens with 7...Be6 and playing for the initiative with the added tempo we'll steal (with either Nc6 or Nd7) from white thanks to their exposed queen on e5. Might be perfectly fine for a blitz game, but otherwise white stands clearly better.
2nd and better option after 7.Qxe5 is: Trading queens on e5 and playing the endgame with a lead in development which partially compensates for being a pawn down, additional compensation lies in the e5 being somewhat weak as it is easy for black to target all forces on it. Clearly, white has a small edge, but that is expected from the line starting from 3...Nxe4 as it is not black's best reply in the Petroff. 

That last point bring us to the mainline in the Petroff, which differs from your move order from move 3 onward: The main move for black after 3.Nxe5 is 3...d6 (played thousands of times more often than 3...Nxe4), which simply kicks the knight back to f3 and then black captures the pawn on e4. Here's a diagram to showcase one typical continuation:
 [title "One possible mainline with 3...d6"]
 [fen "rnbqkb1r/pppp1ppp/5n2/4N3/4P3/8/PPPP1PPP/RNBQKB1R b KQkq - 0 3"]

 3...d6 4.Nf3 Nxe4 5.Qe2 Qe7 6.d3 Nf6 7.Bg5 Qxe2+ 8.Be2 Be7

The difference with the line of 4...Nxe4 is the fact that after Qe2 pinning the knight black can simply reciprocate with Qe7 unpinning and maintaining material equality. Whereas in the previous line (with 4...Nxe4) the trade of knights is forced and white's in time to play d4 winning our d6 pawn after 4.Qe2 Qe7 5.Qxe4 d6 6.d4 dxe5 7.dxe5.
All in all, in the Petroff defense the queens tend to come off the board rather quickly, therefore, it all very quickly transitions into mid-endgame type of position, which requires very technical play (and as such generally not fun). So I definitely encourage you to explore other defensive systems for black against 1.e4, e.g. the Sicilian or the Caro-Kann defense. 
To finish on a fun note, the very line you're interested in is something GM Yasser Seirawan fell for when he was starting out his chess career (around age of 12), where he had come up with a plan: To replicate all opponent's moves and to catch them as soon as they make a mistake, and in the case of the Petroff, as you've experienced it yourself, it ended up costing him his queen of course... :) Here's the story told by himself (Petroff line is around 6:30 min mark), I encourage you to watch the whole thing as it is quite funny and instructive.

user15223: Do not play 3...Nxe4 in response to 3. Nxe5.
I feel this important aspect is missing in the other answers. You can see the primary reason it doesn't work in your game - you end up losing the queen. Given that you can't just move the knight after 4. Qe2, your next-best option is to play 4...Qe7 5. Qxe4 d6 (or ...f6), which still results in you losing a pawn. It's not a disaster because Black does have some compensation in the form of superior development, but Black's position is still undoubtedly inferior.
Avoiding this line is the main line of the Petroff, and is much more stable. The main line goes 1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5.d4 d5. Black has gained a tempo (as seen from his advanced knight), but the knight is somewhat exposed and can potentially be forced to retreat, in which case White is up two tempi. In practice, White can usually achieve this, but needs to play pawn breaks like c4, which balances out. The Petroff is thought to be drawish for a reason.

15543: <p>I recently played a game as Black. My opponent and I transposed into, according to Lichess, the Russian Game, Damiano Variation, when White played 4. Qe2. I wasn’t sure how to reply, and I ended up losing my queen and soon resigned.</p>
<pre><code>[FEN &quot;&quot;]

1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nxe4 4. Qe2 Nf6 5. Nc6+ Be7 6. Nxd8 Kxd8
</code></pre>
<p>As Black, how should one respond to 4. Qe2 by White in the Russian Game, Damiano Variation?</p>
 19027: <p>In Damiano variant you respond 4. ... Qe7 to 4. Qe2. Yes, white takes the knight faster than you, but then you simply make d6 move, and if white knight retreats, white loses Queen. So to black's 5. ... d6 white have to respond with 6. d4, then: you take knight with the pawn and white makes a move dxe5. After those there's a bunch of possibilities that are better to be all checked on lichess.org via its analysis function.</p>

<p>Not to create any complications, you may not beat the e4 pawn after white knight grabs e5. Instead 3. ... d6 is a much better move, in my opinion, making white's knight retreat immediately, after which you can grab 4. ... Nxe4 without a thought after 4. Nf3, for example. And here to 5. Qe2 you simply respond 5. ... Qe7 and thus are able to retreat your knight next move without having a hidden check.</p>
 3594: <p><em>This part of the answer pertains to the first version of the post, where (due to a typo in setting up the diagram in OP) the move <code>4...f6</code> was played:</em></p>

<p>It started to go wrong for you from the move <code>4...f6,</code> which just walks right into <code>Qh5+</code> completely winning for white:</p>

<pre><code> [title "why 4...f6 is a mistake"]
 [fen "rnbqkb1r/pppp2pp/5p2/4N3/4n3/8/PPPPQPPP/RNB1KB1R w KQkq - 0 5"]

 5.Qh5+ g6 6.Nxg6 hxg6 7.Qxg6+ Ke7 8.Qxe4+ Kf7 9.Bc4+ d5 {forced to sac the pawn so Qg4 can be prevented} (9...Kg7 10.Qg4+ Kh6 11.d4+ Kh7 12.Bd3+ f5 13.Bxf5#) 10.Bxd5+ Kg7 11.Bxb7 Bxb7 12.Qxb7 Nd7 13.O-O {and white is completely winning, both on position and material count.} 
</code></pre>

<p><strong>General advice:</strong> Particularly against <code>1.e4</code> systems, you almost always should try to avoid moving your <code>f</code> pawn early on, as white's queen and light square bishop are activated early on and therefore kingside light square weaknesses can be easily exploited. Typical to the Russian/Petroff move order, is rather <code>d6</code> instead of <code>f6</code> in order to kick the knight back. Though <code>4...d6</code> happens to also have been a poor choice in your game, as in that specific variation <code>4...Qe7</code> is really the only reasonable move that maintains the balance of the position. So let's look at that next:</p>

<hr>

<p><em>The remaining part of the answer from here tackles the latest and corrected move order provided by the OP, with <code>4.Qe2 Qe7 5.Qxe4</code> and without <code>f6:</code></em></p>

<pre><code> [title "4...Qe7: cover your king and set up a pin on e5 knight"]
 [fen "rnbqkb1r/pppp1ppp/8/4N3/4n3/8/PPPPQPPP/RNB1KB1R b KQkq - 1 4"]

 4...Qe7 5.Qxe4 d6 6.d4 dxe5 7.dxe5 Nc6 8.Bb5 Bd7 9.Nc3 O-O-O
</code></pre>

<p>And black has a much more reasonable play than they did in the aformentioned line.</p>

<p>Generally speaking, inherent to this opening's move order, moves such as <code>Qe2</code> and <code>Qe7</code> come quite naturally as the <code>e</code> file is opening up quickly and the discovery threats have to be parried, and these queen moves allow you to do so with tempo. Moreover, as you can see from the last line we looked at, trades are abound in this variation, and as black, you should not worry about dropping a pawn if you manage to compensate with development and piece activity (specially in short time control games). By sacrificing the <code>d6</code> pawn on <code>e5</code>, we found a target pawn on <code>e5</code> to coordinate against, which resolves your development plan (in contrast to white's play, hesitating as to whether defend the pawn or to regain initiative), and you have a slight initiative after the long castles. </p>

<hr>

<h2><strong>Addendum:</strong></h2>

<p>Upon further requests in the comments regarding the latest move order provided by the OP, here I provide additional explanations.</p>

<p><strong>Important summary:</strong> In the Petroff, if you want to take white's pawn immediately after <code>3.Nxe5</code> then you should definitely expect <code>4.Qe2</code> as it is white's best move and black has really only one playable reply and that is <code>4...Qe7,</code> as any other move either loses a piece or a pawn. For example the seemingly natural <code>4...d5</code> loses a pawn as follows:</p>

<pre><code> [title "4...d5 leads to a loss of pawn"]
 [fen "rnbqkb1r/pppp1ppp/8/4N3/4n3/8/PPPPQPPP/RNB1KB1R b KQkq - 1 4"]

 4...d5 5.d3 Qe7 6.dxe4 Qxe5 7.exd5
</code></pre>

<p>Therefore, it remains to cover all lines ensuing from <code>4...Qe7</code>, but luckily, white's continuation is also quite limited if they want to play for an edge. The mainline that follows after <code>4...Qe7</code> is already covered in the 2nd diagram of my answer and the discussions thereafter. For completeness, it's worth pointing out that if in the said line (2nd diagram) white captures the pawn on <code>e5</code> with the queen on move <code>7.Qxe5</code>, then black has two choices:</p>

<ol>
<li><p>Avoiding the trade of queens with <code>7...Be6</code> and playing for the initiative with the added tempo we'll steal (with either <code>Nc6</code> or <code>Nd7</code>) from white thanks to their exposed queen on <code>e5.</code> Might be perfectly fine for a blitz game, but otherwise white stands clearly better.</p></li>
<li><p>2nd and better option after <code>7.Qxe5</code> is: Trading queens on <code>e5</code> and playing the endgame with a lead in development which partially compensates for being a pawn down, additional compensation lies in the <code>e5</code> being somewhat weak as it is easy for black to target all forces on it. Clearly, white has a small edge, but that is expected from the line starting from <code>3...Nxe4</code> as it is not black's best reply in the Petroff. </p></li>
</ol>

<p>That last point bring us to the mainline in the Petroff, which differs from your move order from move <code>3</code> onward: The main move for black after <code>3.Nxe5</code> is <code>3...d6</code> (played thousands of times more often than <code>3...Nxe4</code>), which simply kicks the knight back to <code>f3</code> and then black captures the pawn on <code>e4</code>. Here's a diagram to showcase one typical continuation:</p>

<pre><code> [title "One possible mainline with 3...d6"]
 [fen "rnbqkb1r/pppp1ppp/5n2/4N3/4P3/8/PPPP1PPP/RNBQKB1R b KQkq - 0 3"]

 3...d6 4.Nf3 Nxe4 5.Qe2 Qe7 6.d3 Nf6 7.Bg5 Qxe2+ 8.Be2 Be7
</code></pre>

<p>The difference with the line of <code>4...Nxe4</code> is the fact that after <code>Qe2</code> pinning the knight black can simply reciprocate with <code>Qe7</code> unpinning and maintaining material equality. Whereas in the previous line (with <code>4...Nxe4</code>) the trade of knights is forced and white's in time to play <code>d4</code> winning our <code>d6</code> pawn after <code>4.Qe2 Qe7 5.Qxe4 d6 6.d4 dxe5 7.dxe5.</code></p>

<p>All in all, in the Petroff defense the queens tend to come off the board rather quickly, therefore, it all very quickly transitions into mid-endgame type of position, which requires very technical play (and as such generally not fun). So I definitely encourage you to explore other defensive systems for black against <code>1.e4,</code> e.g. the Sicilian or the Caro-Kann defense. </p>

<p>To finish on a fun note, the very line you're interested in is something GM Yasser Seirawan fell for when he was starting out his chess career (around age of 12), where he had come up with a plan: To replicate all opponent's moves and to catch them as soon as they make a mistake, and in the case of the Petroff, as you've experienced it yourself, it ended up costing him his queen of course... :) <a href="https://www.youtube.com/watch?v=rgUQBYOWaOE" rel="nofollow noreferrer">Here's the story told by himself</a> (Petroff line is around 6:30 min mark), I encourage you to watch the whole thing as it is quite funny and instructive.</p>
 15223: <p><strong>Do not play 3...Nxe4 in response to 3. Nxe5</strong>.</p>

<p>I feel this important aspect is missing in the other answers. You can see the primary reason it doesn't work in your game - you end up losing the queen. Given that you can't just move the knight after 4. Qe2, your next-best option is to play 4...Qe7 5. Qxe4 d6 (or ...f6), which still results in you losing a pawn. It's not a disaster because Black does have some compensation in the form of superior development, but Black's position is still undoubtedly inferior.</p>

<p>Avoiding this line is the main line of the Petroff, and is much more stable. The main line goes 1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5.d4 d5. Black has gained a tempo (as seen from his advanced knight), but the knight is somewhat exposed and can potentially be forced to retreat, in which case White is up two tempi. In practice, White can usually achieve this, but needs to play pawn breaks like c4, which balances out. The Petroff is thought to be drawish for a reason.</p>
