Any experienced pet owners familiar with U.S.-Italy travel?

Hey everyone! I’m taking my small dog (8 lbs) to Italy for ~1month and want to confirm I have the process right, and also ask about the export certificate I keep reading about. 

To enter Italy from the U.S., sounds like I just need to get an EU Health Certificate within 10 days before departure, then have it USDA-endorsed. (Microchip and rabies vaccine are good to go.)

I am reading that Italy may require a Pet Export Certificate from ASL when leaving the country? Is this required even if we’re just returning to the U.S. after our stay in Italy? Do you have to get this every time you leave Italy (whether for the U.S. or another country)?

Would love to hear from anyone with experience. Thanks!