Yes they do I can tell you from experience.
nothing
When you beat the elite four.
No you can't use the azure flute after you beat the elite four
You wait until you beat the elite four and professor Elm will give it to you when you beat the elite four.
In heart gold and in gold you have to wait until you get a certain item to get past the monks in the cave, but in silver and soul silver, you can get lugia after you get the blackthorn gym badge, but you can't get Ho-oh until you beat the elite four
no beat the elite four first
The Elite Four becomes stronger by itself once you beat them. All their pokemons become about 20 levels stronger after the first time you beat them :)
Beat the elite four again but they get stronger
After defeating the Elite Four once, their Pokemon are much stronger than before, and with new Pokemon. There is no new Elite Four to battle, though.
Yes you do
You never go to Hoen.
Yes
They are still in the Pokemon league
go to the kanto region
nothing
Finish your Pokedex And Beat The Elite Four once then the second time they have stronger Pokemon
You have to level up your pokemon a lot and make them stronger.