The whole culture vulture concept seems too broad and is kinda silly to me. But maybe you can better explain what you mean by it. I always thought it is when a person imitates a culture for personal gain without crediting or benefiting the originators of the culture. The reasons why I don't think he is a culture vulture under that definitions are:
1) He doesn't just imitate the music. He seems to embrace he musical style and create original works in the same vein. That's different from someone like Timberlake who just hooked up with some black people to get the music and then performed it to build his own career.
2) He always credits blacks and his specific influences for creating the culture and being the reason he has a career.
3) Him surrounding himself with blacks in the band means to some extent he is cognizant of the group that is responsible for the artform and also means that he is benefiting people from the originating group as well as himself even if only on a small scale. And a band is a little different from backup dancers. Backup dancers are just window dressing for performances. They don't really add anything to the music. The band plays an integral part in the music. Hell if you go back to the days of the music that he makes, artists pretty much had to have a good band to get by, so even in that he's being true to the culture.
So what exactly is the problem with him making the music?
The answer to all your questions is "Yes." Sports have associated cultures just like music, and I'm betting you know that. Basketball was most certainly a way of life for the people that created it and grew it into a major sport, just as it is for the people who play it now. The NBA is vast majority black now, and the culture behind basketball in general is a far cry from what it was when whites were the dominant force in the sport, so you could absolutely say that blacks took it away and are now profiteers off of it. Hell, black dominance of basketball is so strong, white Americans can barely even break into the sport anymore. It's not like Bruno Mars' presence is stopping black people from making music anymore.