Enemies reappear if you leave a room and reenter it. It's worth pausing from time to time for a killing frenzy to build up your attack levels. If they dropped a healing item when you killed them before, they'll probably drop it again, which can be useful.
This applies to other things in a room, too – like throwable objects and floating platforms. However, the contents of throwable objects won't regenerate (at least if you already got them).
When you load a save game, or opt to "Continue" when you die, you'll be returned to the start of the current stage with full health. So if you enter a new stage with low health, don't try too hard to stay alive! The other option when you die is "Save". This doesn't mean you load a saved game, but rather save your game (at the beginning of the stage in which you died).
"JA level" and "SA level" in the top right corner of the screen refer to your attacks while jumping and while not jumping. These go up as you use the respective attack. You'll want both to be as high as possible, obviously; it's quite easy to use slash attacks exclusively in the first few stages, then find yourself up against flying enemies with low JA, which can be tedious.