Traditionally (and keep in mind this may have changed when I wasn't looking) the Avengers are funded by the Maria Stark foundation which also handles most of their legal issues. Sometimes Wasp has been known to pitch in here as well.I would think that 'in universe', Tony, Steve, Janet, and Thor (just off the top of my head) would control the name and licensing of the Avengers. I'd think that they would also control any type of usage of the name including: West Coast, Great Lakes, Young, etc. As for why can't some heroes become Avengers, I was about to go into a long rant and realized how intricate the process would actually be to make sure the term 'Avenger' means something and isn't taken for granted. Don't think I can answer this.
Captain America legally owns the role of "Captain America" and was given all rights associated with it by the government. The government has also issued Cap a "Champions License" which means he can legally assemble teams of superhumans for the purpose of crimefighting whenever he wishes. (And this is yet another reason why Civil War never made sense.)
Thor (and sometimes Namor) has also assisted with funding by showing up with arms loads of gold at various points.