Do taxis wait outside of the S. Pietro Station in Rome or is there a taxi stand nearby? Also, can you flag down taxis in Rome? I've seen different rules in various cities in Europe and I I'd like to know what the taxi protocol is in Rome.
There are usually taxi stands outside of every train station and major site. If you need a taxi in Rome, usually the best rate will be at a taxi stand. They're very abundant, and you won't need to worry about getting one. If you need a taxi at your hotel, they will do one of two things the cheapest will be to direct you to the nearest taxi stand to the hotel, or they will call one for you, and the meter will start when they are called. I'm older now, and sometimes I just pay the extra EU for the convenience of having the taxi pull up outside my hotel. No worries you'll get a taxi when you need one.
Train stations almost always have taxis at the taxi stand outside, or one will show up within a few minutes (taxi drivers know the train schedules). If none appears soon, there may be a number for you to call, or a bar can make the call for you (at least buy a coffee). It's really hard to be stranded in Rome, in my experience.
You don't hail taxis in the street in Rome.