And if it "doesn't happen," will it lose HP for the remaining turns? So, if a burn lasts for 3 turns, but only one turn has passed, will HP be deducted for the remaining two turns after the battle? If the burn is "permanent," will it die? Is that correct?

And if you come to a place where a battle is already taking place, can you "join" one of the sides, like Ginza or Lampra? Or are these "tricks" only allowed for NPCs?

Can you obtain ingredients for cooking through looting?