Like the title says. What possible sequence of events could lead to Giordano Bruno, who was as you probably know burned at the stake for heresy in 1600, becoming the Pope? What name would he take? How would he affect the Catholic Church? Could he still be open with his highly unorthodox beliefs about the universe, or would he have to stay "in the closet"?
As his faith is steeped in Arian and pantheistic theology, his becoming pope during the later half of the 16th century is most extremely improbable. It's simply not going to happen.
That said, to specifically answer your questions:
- No plausible, realistic sequence of events could bring this about. Any scenario would require either a radically different RCC or a radically different Bruno. Either way, one or both would be completely different than in OTL and unrecognizable to us.
- Irrelevant, as the chances of the OTL Bruno becoming pope of the OTL Church are between none and nil, and definitely less than nada. However, if he must have a papal name, how about Julian.
- The Church becomes ultra-orthodox as Pope Julian employs extreme measures to prove that he isn't an Arian heretic.
- He would need to keep his highly unorthodox theological views buried very deep within himself.