It has long been thought that the Bard was a Roman Catholic, and rightly so, I believe. Protestants just don't understand human nature so well. The finest newspaper in the Anglophone world, The Telegraph, presents further evidence thereof here.
Certainly, however, Shakespeare was married to Ann Hathaway at the time, so he would not have been a seminarian at the Venerable English College at the time. As to his business in Rome, we can only speculate.