Could you tell me how you record your piano pieces so quickly? When I try to record something I compose, I make so many mistakes that I have to keep practicing the piece for about a week or two before I can record an almost perfect version. Does it just get easier with time?

It's just something that comes with time. I've done 1400+ videos and before YouTube I was playing every day anyway.
If you stick with it there's no doubt that you will be able to just quickly record them =)