Hi Cathryn,
Here's my two-cents worth.
"What makes a sport a sport?"
-- To me, a sport is an activity involving some form of physical exertion which is moderated or governed by rules specific to that activity, and often exercised competitively.
"Sport or Art?"
-- I think they're both. Ballroom dancing is primarily performance art in that it deals with artistic movement, much as ballet, kung-fu, tae kwan do, or any dance or martial art. However, it is also a sport when it is undertaken as a competitive activity (these days under the name DanceSport) much like ice dancing or Judo.
"Ballroom Dancing as an Olympic Sport?"
-- I am rather ambivalent on this point. There are many pluses and minuses to both--and too lengthy to go through here.
"Value add as an Olympic sport?"
-- Any sport could always benefit from media exposure. Ballroom dancing could gain more in terms of prestige and recognition if included in the Olympics. The other side of that question is, "is ballroom dancing ready for prime time?" It could get some media coverage but not much--initially. Whether it would be enough to merit the effort for inclusion into the games so we get more exposure for dancesport remains to be seen. It needs to capture the fancy of the viewing public and be presented in a way that changes their existing perceptions. It would then gain a strong following beyond the existing afficionados.
"How long does it take to master a dance?"
-- Mastery is a life-long undertaking in almost any discipline. To meet baseline at the National level (Open Championship level), I would say eight (8) to ten years. At the International level, perhaps thirteen or more of intense dedicated training.
"Dancer trains as hard as any other athlete? "
-- Yes. I hit the gym five times a week for endurance and strength training. I practice two to six hours per day four to six times a week depending on my work schedule. And compared to top dancers, that is hardly enough.
I hope this helps.