Flying from Rome to DFW: non-stop or connecting in US

Yes, with a connection, at your first landing point in the US, you have to go through Immigration control, then pick up your luggage and walk through the "nothing to declare" exit and then put your luggage back on the belt to be loaded on your flight to DFW.

If you can fly non stop, you are lucky to have that option!

Time? That can be tricky. Both ORD and JFK are busy airports serving several airlines for incoming international flights, so things can get backed up. Since you appear to be flying American Airlines, I'm surprised Charlotte did not come up as an option.

If you will have only carry-on, you save a bit of time by not having to wait for bags, but 2 hours might be a good target? You can download the CBP MPC app (Custom and Border Control Mobile Passport Control) It allows you to enter in all your info as you are taxiing to the gate, then use a separate line (maybe, depending on the moment) that promises to be quicker.

At ORD, you have to change terminals if you are on American or United, which means going through security again. So I for that would have 3 hours for that. Delta is now in the International Terminal and you can walk to your next gate, so 2 hours should be OK.

You will always go through security again because you have had accessed to your checked luggage. I don't know if it is possible to stay behind security with only carry-on. We tend to check everything coming home.

I don't know if it is possible to stay behind security with only carry-on.

You still have to go through security, because even if you don't have a checked bag yourself, you still get routed through the same path through baggage claim as everyone else (there's no "side path" after immigration for carry-on-only folks that takes you back directly to the gates).

you still get routed through the same path through baggage claim as everyone else (there's no "side path" after immigration for carry-on-only folks that takes you back directly to the gates)

True at JFK and ORD, but there are 2 instances I know of where there is a side path. SkyTeam arrivals @ Minneapolis (Delta, KLM, Air France) get a side path directly back into the concourse (with dedicated security just for them), also @ Dulles Star Alliance arrivals (United, Lufthansa) have a similar side path.

You always go through Customs and Immigration at the port of entry into the US.

...at your first landing point in the US, you have to go through Immigration control

I was on a Qantas route coming from Sydney/Fiji with a stop in HNL on the way to LAX, where mainland US destination passengers stayed at the gate and domestic passengers for the mainland boarded, mingling international and domestic passengers onto the same plane. Everyone went thru immigration on arrival at LAX, even the domestic travelers. No idea if this is still going on. Believing that the announcements directing LAX passengers to not go through immigration must be false, I started heading toward immigration but was able to run down an escalator backwards and get back to the gate.

True at JFK and ORD, but there are 2 instances I know of where there
is a side path. SkyTeam arrivals @ Minneapolis (Delta, KLM, Air
France) get a side path directly back into the concourse (with
dedicated security just for them), also @ Dulles Star Alliance
arrivals (United, Lufthansa) have a similar side path.

Do these avoid security though? To be clear by "side path," I meant a path directly after immigration that completely bypasses the baggage claim area, which would in theory allow those passengers to avoid the security re-screening.

There's dedicated security staff and machines at this point just for these passengers to allow them to stay on the same level and spill directly into the gate area, both locations (MSP/IAD). Security is temporarily staffed just for arrivals. I only mention it because it could be a factor when someone considers a tight connection (note it's airline specific).

Unless there is a big, big difference in the cost, fly direct to DFW. You'll not have to worry about connections, going through security again, and retreiving your bags. Less stress all around. Why be stressed at the end of a great trip?

What TC said.