Question:
When you're flying on a Carpet of Flying and do something that requires a Fly check, do you make it with your stats or does the carpet make it with whatever it has in the Fly skill?
It doesn't have its own Fly skill. The item references the Overland Flight spell, mentions that it can hover without a Fly skill check, and grants a +5 bonus to other Fly skill checks. Since the carpet has no Fly skill of its own, it must be conferring bonuses Fly skill checks made by its "driver".
I can't really think of any time a carpet would have to make a fly check. Most of the fly check things apply to creatures, not objects.
A person standing on a carpet isn't flying, so those checks wouldn't apply to them either.
Unless your flying carpet is an intelligent item AKA aladin's magic carpet.
In that case I guess the carpet would have its own skill.