it depends on what you consider the number of sports/teams. VT has stated over the years that it has 21 sports teams. If you go to Hokiesports.com the number of current sports listed in the "Sports" dropdown does equal 18 (excluding women's golf). However, to arrive at the way they count the total number of "sports" is you have to add women's cross country, women's swimming and women's diving, thus adding three more to get to 21. With women's golf it will be 22.
Yes. With some of the sports you have to drill down to determine if they have men's and women's. If you look at the SEC website, it looks like they have soccer. If you drill down, you find out that it is only women's soccer. They do not have men's.