Yes, 3 minutes for AP, 20 minutes for BP. The timer resets when doing an action that is listed to cost that resource (even if the cost is zero).
So for AP, the timer resets when you start a quest (I think when the loading starts) or a harem episode. For BP, I think it's when you start loading into a raid. But I'm not 100% sure when it comes to a situation where you spend a half-elixir or seeds and then back out.

The +1 AP at rank up continues until rank 60 (at which point you should have 80 AP). Then it becomes +1 AP for every two ranks up until rank 104 (at which point you should have 102 AP). Then the rate slows down again, but I'm not sure on the exact rate just yet. I'm at rank 111 with 104 AP.