If I run out of Pokeballs, does that mean I can't battle them again?

Defeat the Elite Four again, and it will re-spawn.

No, you could reload the nearest slot of game, which has magma stone and defeated elite 4.

Either stock up on pokeballs, or weaken the Pokémon more before you try to capture them to raise the chance of capture.

No, but if you saved the game, you may do it again with a soft reset.

A quick ball and ultra ball should suffice. Don't use a master ball, save it for Mewtwo.

It disappears from the game. Knocking out Zapdos, Articuno, Moltres, or Mewtwo removes them from the game completely. That's why you should save your game before trying to catch it,...

It honestly depends. You could consider the Pokemon your first encounter, and catch another after the mandatory knockout or ignore it and catch something else when you can get some...

On a sheet of paper, write down all of the negative things about her you can think of. Every time you miss her, take out this sheet of paper and...

It is, but you have to give it a status effect first. Sleep and freeze is the most helpful (though you have to get lucky to get a freeze). Without...

Once you have played it once you can't do it again unless you die and restart.