I have a tiny soft skin "growth" on my labia majora close to the fold that connects to my inner thigh, the bump has a pubic hair growing through it (not exactly the center) can it be a genital wart?

More than likely not: the hair is probably ingrown. A wart would be more fleshy and would get bigger over time. You can always have it looked at. I would be more concerned if it increased in size.

Growth with hair: More likely is a skin tag. usually they have a stalk connecting them to the skin and they do grow in areas near folds. a doctor can remove then either by cutting or cauterized. They are usually benign the Birt-Hogg-Dube syndrome is very very rare. F/U with doc http://emedicine.medscape.com/article/1060373-overview