Is this considered a "variation" of Scholar's Mate?
Scholar's mate playing black

I guess so. But don't play like that if you care whether you win or lose. What was the plan if Bxg5 instead of Nxg5?

Yap you are right Bxg5 gives him a piece and advantage. Also in general I have noticed when I move my queen out early against decent/good players it creates problems for me.

Yeah it isn't a good idea to bring the queen out early. It just becomes a target and the other guy can develop pieces whilst attacking it. A general rule - knights and bishops first, castling, then the heavy pieces. Think of it as a race to develop your pieces to a useful square. Can your stuff be attacked where you put it? If you win this race you have an advantage because you are effectively playing with more pieces so you can do more things.
I am ranked so low I was actually able to successfully do a Scholar's Mate playing black ( :
I's this technically considered a Scholar's Mate?