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The current fixed calendar works in such a way so that the first day of Rosh Hashana can never fall out on a Friday, but this is the very day on which the first Rosh Hashana fell! What gives?

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The reason it can't fall on Friday or Wednesday AFAIK is so that Yom Kippur won't fall on Sunday or Friday and there would be two days in a row without the ability to cook. Is that the kind of answer you are looking for? It doesn't directly address your concern, but presumably it was just a cost-benefit analysis on the part of the calendar setters. –  Double AA Sep 19 '12 at 4:52
The first Shavuos also fell on a Shabbos, while according to the rule of לא בדו פסח and א"ת ב"ש, Shavous can never fall on a Shabbos. –  Michoel Sep 19 '12 at 4:55
@DoubleAA if the answer is that the calendar was fixed at a certain point but not prior, then that's the answer, but how can the anniversary of something be "fixed"? Shouldn't it always fall on the (lunar year) anniversary of the event? –  yoel Sep 19 '12 at 4:58
@yoel That's the beauty of Kiddush haChodesh: we define when the lunar anniversary is! You can ask the same question about birthdays, eg. because beis din declared it a leap year, I'm no longer bar mitzva? –  Double AA Sep 19 '12 at 5:01
@DoubleAA like the famous Yerushalmi: "בת שלש ויום אחד ונמלכין ב"ד לעוברו הבתולה חוזרין –  HodofHod Sep 19 '12 at 5:52

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The Talmud in Rosh Hashanah (20A) explains that Yom Kippur shouldn't fall out on the day before or after Shabbat, since two consecutive Shabbat-like days could be problematic. If Rosh Hashanah were to fall out on Friday, Yom Kippur would fall out on sunday. Two reasons are given (see the gemara there for a discussion of these two reasons):

  • If someone dies on the first day, the body would not be able to be buried until after the second day.
  • Vegetables (which were eaten raw) wouldn't survive the two days.

I would argue that these reasons weren't really applicable the day man was created, since there was no death in the world yet (until Adam sinned).

It would be interesting to see what Adam's observance of Yom Kippur would have been like, had he not sinned. Would he even fast? Would it still be day of Atonement?

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