All in the question.
...and that's it. The Pokemon's levels don't increase, and you don't unlock anything new.
The same thing happens when you beat the E4 the first time. Except you aren't given the train ticket or battle AZ. You just go stand in your bedroom. Hope I helped!
Nothing; there's no team increase or level increase. They have the same teams and give out the same money. They're just high level trainers you can battle.
you'll gain experience and money like usual and will have another Pokemon champion record. you'll get to watch the credits again and end up back in your house. You also will get some of the legends to respawn if you accidentally made them faint or ran away from them, this may have limited uses though so it's best to just save right before starting the fight with the legendary and resetting if necessary.