I graduated from BYU in Ballroom Dance. It's a great school.
UVSC is second only to BYU.
LDS issues at BYU:
*There are indeed ballroom students who come there who aren't LDS.
*Tuition is higher if not LDS, like out-of-state tuition, but still lower than many state schools. (Why is tuition higher? Faithful LDS members donate 10% of their income to the Church and the Church uses a part of those funds to offset LDS tuition).
*You must agree to live by Mormon moral standards while attending BYU: No alcohol, no smoking, no un-wed sex. If you fudge, you're expelled; permanently, I believe.
*Note: There are no polygamists in Provo or Orem. (Or if there really are, they hide so well you'd have a better chance winning the lottery than meeting one. I lived there for 7 years and never so much as heard of one.) The polygamists everyone likes to rave about are excommunicated, rogue Mormons living in remote parts of southern Utah.
Career/degree considerations:
*You need no degree whatsoever, nor will a degree aid you in professional teaching at a studio. They are interested in skills only: selling, teaching, dancing (in that order).
*To teach dance in High School you need a teaching certificate, not a dance degree; though that may be helpful.
*To teach dance in college you need a masters. BYU, I believe, does not currently offer a Masters in dance; though a bachelors with Ballroom emphasis or a Ballroom minor is available.
*To compete successfully, you will gain much more from private lessons and practice than you will from college classes. But college classes are a much cheaper way to build the basic skills.
*BYU has the largest and best college program in the US, followed by UVSC. There is also a lot of performance opportunity with the Ballroom Dance performing teams. The rest of the Dance program is, as a whole, among the best in the US and extremely well-rounded (though many schools surpass them in Ballet or Modern technique).
*Dancing in college will place you in a fun community. If you are looking for a social ballroom atmosphere, BYU or UVSC definitely offer that opportunity
Good luck.