Durham is pretty liberal, and while most of the Vegan/vegetarian place are in Chapel Hill, most Durham places have at least a few Vegan friendly menu items. Duke is also pretty active about offering vegan and vegetarian friendly food options.
Don't think that Durham is Deep South. It's not. It's more East Coast than South. Outside of the Triangle area, however...
Durham is semi-bikable. I know lots of people who bike. It's hilly, though, and there aren't many bike lanes. There is, however, a thriving bike culture.
There are State and National parks all over North Carolina. You're a few hours from both the beach and mountians. Check out the Eno River. It's incredible!
Duke hosts the American Dance Festival every summer. http://www.americandancefestival.org/ Though from what I gather, many dancers are actually in Greensboro.