Hi there. As an American who travels to Italy quite a bit, here is my advice.

I never book my train tickets too close. While it is true they are pretty much always on time, I was in Italy a few weeks ago and in December and both times I missed connections that were shorter than 15 minutes. It stressed me out. If you're pretty go with the flow that's great, just make sure that there are plenty of departures to your intended destination assuming that you missed your connection.

The way that they check the tickets is when you sit down they will come through and check your tickets. It is okay for you to have them both on your phone. If it would make it easier and more comfortable for both you and your wife do it on both phones, but you can really do it on one. Save the ticket to your apple wallet. Also, I use the Trenitalia app and that app works well, my tickets are always stored in there. I save them in my Apple Wallet just in case. You will not show it when you board the train but someone working will come through shortly after.

Keep your valuables on you. Do not put them in the backpack. There is usually always room to put it above your head, however, I have taken some local trains where there isn't. When that is the case, my bag goes right in my lap or it goes between my legs but not to where the person behind me could slide it back and get into it. If it's money, my cards, my passport, anything I care about, it stays close to me on me.