I have an hour and 20 minutes layover in Frankfurt, do I have enough time to get on my Rome flight and do I have to recheck my bag?

you'll need to provide more details.

- have you already purchased your tickets, or are you asking this before you purchase so you don't make a bad mistake?
- is this on a single ticket or two tickets purchased separately?
- what airlines for each leg?

If both legs are on a single ticket and the airlines are UA/Lufthansa (LH) or LH/LH then
- you don't need to claim and receck your bags
- yes, 80minutes is ok assuming your incoming flight is on time.

In my experience, if you miss your connection at FRA LH will put you on the next flight to Rome. ymmv.

If these are separate tickets, but still on UA/LH then chances are good (but not guaranteed) that they'll check you bags all the way to Rome and you'll be ok.

If these are on separate airlines/alliances and you need to claim then recheck you bags - then your chances of making the baggage claim cutoff time (which is likely 60minutes), is nil....

You will have to go through passport control in Frankfurt, as that is your first entry into the Schengen zone.

I have connected thru Frankfurt numerous times, although the airport is very large it is well organized. You should have no problem connecting. Unless your incoming flight to Frankfurt is delayed you will be fine. The airlines research these connections and allot the appropriate time to gain your connecting flight. A few years ago we had a flight from Boston to Paris delayed which resulted in our missing our connecting flight to Berlin. The next flight to Berlin was a little over four (4) hours. Unbeknownst to us there is a provision shared by the European Union airlines which compensates for any missed flight over 3 hours if the airline is at fault due to a late arrival (we were on schedule as we arrived near Paris but then had to circle in the air for over an hour for reasons we never found out). As we scrambled to find a flight to Berlin, Air France advised us of the aforementioned provision. For missing our connecting flight me and my wife each received $300.00 (approximately 2 weeks later) a refund for the train connection/reservation we missed in Berlin (to Heidelberg) (approximately $80) and a voucher for $40 each for the airport restaurant's during our 4 hour delay. We had a great breakfast, some coffee's, relaxed and put $600 in our pockets. We made it to Heidelberg several hours later than we planned, but it was well worth it.

Some European airports use a central scanning post to check carry-on bags. The possible line-up could delay access to the next flight.

In my experience, no, 1hr 20 is not enough time.
We had a single ticket for both legs of our trip (Milan to Toronto via Frankfurt with 1 1/2 hr layover) and the flights/ times were approved/bundled by the airline. We were delayed by 20 minutes out of Milan and waited on the tarmac in Frankfurt for maybe 20 minutes. Our bags were checked through and so when we made it into the airport we literally ran (the gates were far apart). We had to pass through a checkpoint (can’t remember why since we were already in EU) where there was a long wait. We made it to the gate a minute after the doors closed- so off to the Lufthansa desk to be reticketed (along with several others from our flight). There were no more flights available for days so we spent the night in a hotel (paid by Lufthansa) and then were routed to Brussels and then Toronto. In Brussels we were trapped in another long passport line (again, why?) and almost missed the flight to Toronto again!
If you haven’t yet made your flight selection, if it were me, I would find something with at least 3hours between legs.

You need to go through passport control to exit Schengen as well. Otherwise people could overstay the 90 day limit without being discovered.