Raid battle pass

Yes, you can pick up a pass on day 1, use it on day 2, then spin a gym to grab day 2's pass. You can only carry one free pass at a time.

Yes. As Skywalkered said, you get a free raid pass when you spin a gym if:

A) You don't have one right now, and
B) You haven't gotten one today yet.

I used to spend yesterday's raid pass, and then leave the battle, spin the gym to get today's free pass, and then do the battle - that was back when you couldn't spin the gym after winning the raid, and I didn't want to have to go to another gym to get my daily free pass. Of course, can only do that once a day, after that you don't get the free pass, gotta wait until tomorrow.