Regarding rest days: I could be wrong, but I say do what works for you. Listen to your body.
I don't really know which is better (wish I could be of more help!) when I was working on my pushup challenge, I liked that the amount varied vs steadily increasing. Not sure there is a "better."
Not sure about the part about increasing muscle.
I'm sorry I wasn't more help! I just wanted to try and offer a little advice...since you've been waiting for an answer.