The only things that happen after you beat Red are that you are given the Kanto and Hoenn starters from Oak (in his lab) and Steven Stone (in Silph co.) Otherwise, congratulations! You just beat the hardest trainer in the game... and showed Ash Kechum who's boss.
nothing
With Pokemon.
Nothing.thats it
Pikachu
Nothing...I think you might be able to obtain Kyogre but I don't recall anything else :)
go to prefacer.oke then you get one stear pokemon
You finish Pokemon heart gold after you beat red.
beat the Pokemon luegue then go to the Pokemon centre
Yes
They are still in the Pokemon league
Yes.
you gp to seafoam islands