They can surely practice a sport, and be good at it. But to make it to the Olympics, you also need a support base: sponsors, trainers and their support staff, a place to train. I don't see the Vatican having all that infrastructure, although they could outsource that part to some Italian establishment. But first, they need to be lucky to have someone in their support staff (I doubt a 50 years old cardinal could qualify, no matter how gifted) good enough in a particular sport, young enough, willing to change his citizenship and have the upper clergy support.
It's not impossible, but those are long odds.
Not at all! Every nation who participates in the Olympics are guaranteed a number of quota spaces. I don't know the exact math, and quota spaces are limited to certain sports- swimming and athletics for sure, possibly others? Next games, try to catch a preliminary heat of the 100 meter dash at 6 AM local time and you'll see performances that would barely impress an NCAA division II coach.
So if you somehow got the Vatican interested in participating, they would be guaranteed a certain number of spots. Probably just one, given their likely population, maybe more if we're somehow talking all Catholic clergy.