Just some random answers to questions I noted earlier:
Power Bounce is not infinite against any enemy, it is related to the power of the enemy (I think) i.e. you can Power Bounce 70 times on a goomba, only 4 or 5 times on Bowser. You can tell whether or not power bounce will keep working by equipping the "all-or-nothing" badge which makes you miss if you miss an action command. If you don't miss on Mario's last bounce, that means it was the last allowed bounce. If you do miss, it means you just screwed up.
HP does not need to be upgraded. Find out what badges you're going to use for BP and then pump FP. I'm not positive but I think without any outside leveling up you get to at least level 10 by the end of the game; possibly more, that's just a guess. This should be more than sufficient.
Close call works if Mario's HP<5 which is pretty easy to get to with only 10 HP. I'm not sure if it can be manipulated to work every time but it seems like it can.
The power badge is almost certainly not worth it, except maybe at the very end. I question the use of the Mega Rush badge. It takes about 30 seconds in exchange for saving a few turns of varying lengths of time in about a dozen battles. While this would almost certainly make up for the time on its own, you have to take into account the fact that Mario needs to be brought to 1 HP before, or during each of these battles which is time-consuming on its own and may even be impossible or require preparations that take even more time. The only way to be certain would be to do the game both ways. I'm not sure which would be faster, I'm just saying that Mega Rush is not guaranteed to be faster.
I think upgrading Bow is a good idea; she does 5 damage with a free attack although it takes more time than some of the other characters. That one extra free damage may be key though. There are some upgrade blocks that are not out of the way at all including one in the chapter you get Bow. In particular she is great against the Jr. Troopa that you fight right after the chapter you get her. Other than that I would upgrade Goombario twice, if possible, because he does a lot of damage fast, and Watt twice because she is useful against anything with defense, especially final Bowser.
I don't think there is a fire attack badge, but there is an ice attack badge and I don't think it's out of the way. You will actually need the ice attack badge to Power Bounce on Lava Pirahna.