The use of got free does not sound natural to me (American English).
I suggest using became free .
I would also use ago :
The bathroom became free only 10 minutes ago.
As for The bathroom was vacant only since 10 minutes ago.
This doesn't sound natural either.
Here I would use the present perfect:
The bathroom has been free/vacant for (only) ten minutes.
ell.stackexchange.com/users/29924/Alan Carmack
The bathroom was vacant only 10 minutes ago was indicates a state.
This means that 10 minutes ago the bathroom was vacant.
The only gives the impression that it might have been occupied before or since, but the sentence does not say.
It might have been vacant before, it might have been vacant after that, it might still be vacant now.
The bathroom became vacant only 10 minutes ago became indicates a change of state.
This means that the bathroom was occupied until ten minutes ago, and then it became vacant.
It does not specify whether it is vacant now.
back is also possible, but according to this NGram , ago is much more widely used.
ell.stackexchange.com/users/30978/JavaLatte
You can use "got free" in the sense of "Reach or cause to reach a specified state or condition:" ( http://www.oxforddictionaries.com/definition/english/get #3) in a sentence like "
The attackers tied up the hostages, but they got free 10 minutes ago."
But that sense of "got" doesn't work in the OP's sentence, because the bathroom didn't take any action to remove the people who exited from it 10 minutes ago.
In the OP's second sentence, "since" may be a wrong translation of a word like "depuis" in French, which is translated as "since" or "for" in English depending on the context.
"
Il habite à Paris depuis 1990" =
He has been living in Paris since 1990.
"
Il habite à Paris depuis dix ans" =
He has been living in Paris for 10 years.
(Note: French uses the present tense of the verb in those examples, but English does not.)
"The bathroom has been vacant for 10 minutes" would be the commonest usage, but "The bathroom has been vacant since 10 minutes ago
" is just about possible - a sentence like "The bathroom has been vacant since before lunch" seems perfectly OK (in British English, at least).
ell.stackexchange.com/users/19004/alephzero
I would tend to say this phrase as: The bathroom was freed up only 10 minutes ago.
or (changing the phrasing and subtext only slightly) The bathroom has been free for 10 minutes.
ell.stackexchange.com/users/40307/Ian MacDonald
[I would post this as a comment, but no reputation]
I think that something like >
The bathroom only opened up 10 minutes ago.
(I don't feel good about this option anymore)
The bathroom finally opened up 10 minutes ago.
might be what you want.
It changes the meaning a bit though, so I'm not sure.
EDIT: I'll try to explain.
Using "finally" emphasizes that the bathroom was occupied for a long time.
Whether you use "finally opened up 10 minutes ago" or "opened up only 10 minutes ago" depends on whether you're in a situation like #1 or #2:
You're saying that the bathroom was occupied for a long time, and it just now finally opened up.
Maybe in a conversation like this: Me: "Is the bathroom still occupied?"
You: "
No, it finally opened up 10 minutes ago."
You're saying that the bathroom has been open for a short time; you're emphasizing that it has been a short time.
Maybe in a conversation like this.
You: "Bathroom's open."
Me:
"When did it open up?
Has it been open for a long time?"
You: "
No, it opened up only 10 minutes ago."
ell.stackexchange.com/users/40322/xmq
The bathroom got free only 10 minutes back.   Is 'got free' the right way to say it? What about    The bathroom was vacant only since 10 minutes ago.
Is it right to say, "The washroom got free only 10 minutes back"?
ell.stackexchange.com
<sentence-construction>
ell.stackexchange.com/questions/100593
ell.stackexchange.com/users/37548/M. Vohra
The use of  "got free"  does not sound natural.   You could use "became free". Alternatively, you could say "the bathroom  was freed up  only 10 minutes ago".  "Got free" does not sound natural to native speakers, they would most likely use another phrase like "became free". "The bathroom was freed up" is another alternative.
"Got free" does not sound natural to native speakers. "The bathroom became free" is something a native speaker would use. "The bathroom was freed up" is another alternative.