You're right, there isn't generally a quick and easy Q&A for travel and many things you just learn along the way as you're doing them. So I can answer from personal experience from the many trips I've taken back and forth between the US and Europe and from living here now for over 12 years.
All of the EU countries (except for the UK) are also part of the Schengen area, which allows people to move across borders freely without the need for visas. This is for EU citizens, of course. But if you don't need a visa to visit the EU based on your citizenship, then all you're asked is to show your passport at the first place you land in. In your case, it will be in Paris.... and if from there you catch another plane to Italy, you won't have to show your passport again once you arrive here because you're already in the Schengen area!
So I can confirm no visa is required in Paris either for US citizens. And no European country has ever asked me to see proof of immunizations, ever!
Ah, that hour you think you have in Paris? It will barely be enough to make it from your plane, through passport control, to the shuttle bus to get to the next terminal, maybe an extra line of security and your plane will be boarding. So don't think you have lots of time in Paris, you really won't - be quick to move around so that you don't lose the connecting flight!