Where does Zapdos go if I run out of Pokeballs or faint and am not able to catch it?

As long as your visa is in order (assuming you are from another country originally), you should still be able to. The law only prevents unlawful residents from obtaining handguns,...

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

Yes, which is why you should only battle them once you think you have a good amount of Pokeballs.

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

It's that dreaded low catch rate of 3. You just gotta keep trying until you catch it.

Because they have to be either paralyzed or asleep. The Pokémon has to have some sort of status, so the Pokeball doesn't miss.

It is, but the Zapdos has to have low HP.

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...

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...

No, it doesn't. However, do you have an Xbox 360? if so, try there and then it should work, provided you downloaded it.