Given the recent seismic activity, would it be prudent to have all pertinent papers--and maybe money too--with us at all times--day trips, out to dinner, etc.?
If that is what you do every time you travel to the West coast (California, Oregon, Washington) then do. The West Coast is seismically active and since the seismic events are potentially proportional to the length of the fault line, the San Andreas fault is capable of producing earthquakes much more powerful than the ones in Italy. The chances of a destructive earthquakes in the areas you are visiting are nearly nil.
We do wear chest pouches and always carry our passports but I'm thinking that just having images of this stuff on a flash drive might not be the best thing in a power outage.
Images of what stuff? I only have photocopies of my passport, because I leave the actual passport in the hotel safe (or at home).
And that's another thing--we've been schooled on past trips to not carry tons of Euro around, that there are ATM's everywhere. That's fine & well--except when there's no power. I'm thinking we might want to have some extra Euro while walking around--not crazy amounts, mind you--but a comfortable amount.
I always have at least 200-300€ in my wallet. But I always have that much in US$ in America as well. I am a pickpocket's dream. However I've never been pick pocketed, and I've never known any of my friends and relatives in Italy who has. Therefore one must safely assume that it happens very rarely. Over 90% of wallet pick pockets happen on crowded public transportation (city buses and subways especially). If you are extra careful when you are aboard those, you will be fine (in your case you will probably use city transit only in Rome).