Since you have never flown internationally, let's break it down.
1. Immigration is what clears people to cross a border. Customs is what clears luggage (or other "stuff") to cross a border.
2. Traveling to Italy, you will clear immigration via a passport check at your first stop in Europe. You will clear customs at your final destination. If you are flying straight from the US into Rome and that's it, both will be done at the Rome airport.
3. Customs in most European airports is a non-event; it's up to you to declare if you are bringing in anything that exceeds your customs allowance (highly unlikely, as a routine tourist) and if not, you just get your bags and leave the airport.
4. Returning to the US, you clear both immigration and customs at your first arrival point into the US, which would be DFW for you. You clear immigration, then collect your bags, clear customs, recheck your bags at a post-customs luggage drop-off spot, then go through security and on to your connecting gate.
5. Having global entry means nothing when traveling to Europe. It's a US program. But it will expedite the process when you return to the states, because you will follow signs to GE kiosks at DFW and usually much shorter lines for them. But as long as your GE number (known traveler number) is entered into your flight reservation, you won't actually need your GE card. When you scan your passport at the GE kiosk, it will reflect your GE status.
Hope this helps!
