I think VT is about as passionate as any program in the SEC (when they're winning). It's a big pissing contest on who has "the most passionate fans", but even Ole Miss, Miss State, and USC fans are pretty passionate. Just because Alabama is the big dog, that doesn't mean UF, UGA, Auburn, or A&M fans are less passionate. VT fans are just the same.
I think financially VT would jump at it. They're the step child. I also think NC State would jump at it. The "tobacco road" schools (Duke, UNC, and Wake) treat NC State like a step child too. UVA turns their nose up at VT, mostly due to academics. I've been to a UVA game (one of my close friends played for Bronco Mendenhall at Oregon State - he was his position coach). That's not passion! VT is right down the road from Knoxville and probably closer to some SEC schools than UF is. I think they'd be a great fit.
Yes, UVA has a solid baseball and basketball program (although the basketball has been pretty dormant until the last few years).