1) Ap is 1 per 3 minutes, and BP is 1 per 20 minutes.
2) Don't think so - I've used a BP restoring item, gone into a battle, came out, and had 1 more than i did 3 minutes ago. That being said, if you restore to MAX, then I'm sure the timer would be reset.

Others can confirm or deny this - but i'm pretty sure.