The league itself being tax exempt is so that the teams don't get double taxed (at the league level and the team level) and is very sensible.
Goodell's pay ultimately comes from the owners' pockets. So who cares what they decide to pay him. It's not like no taxes are paid on the profits from nfl football.
Not all the stadia are publicly funded. To the extent they are, teams generally (there may be exceptions) pay a fee for use.