Yes they do, I can't find any info of them though. Their wikki page only states that they represent Jamaica in rugby sevens. Since Their 15's is ranked 84 in the world and they only play a couple of games a year. So probably its just the same players playing both

Yes they do. They compete in qualifying competitions for all the main tour events but have never got through to date. They are aiming to compete at the Rio de Janiero olympics in 2016