I believe bottarga is allowed, as are most (if not all) salty cured and or canned fish. No meat.
Many cheese are allowed, and I always get them vacuum packed. Roscioli is my favorite place in Rome, right on the Campo for that.
I like bringing back super high-quality anchovies in tins, tuna in tins, etc. etc. and exploring one or two groceries' canned and packaged food aisles. I always try to find a half pound of high quality dried porcini at like 1/4 of the price here.
Explore the Testaccio Market for some goodies for your suitcase.
But - may I ask where you're located? Cause everything on your list is pretty much available here in the US.
This page may be helpful to you...Customs and Border Protection
QuoteUSDA’s Animal and Plant Health Inspection Service does not regulate the importation of most seafood items. We only regulate breaded seafood products, and only if the breading contains animal-origin ingredients (milk, eggs). Please contact us directly about breaded seafood items
And I'll add - No fruits or vegetables. No live chickens.
If you drink, stop in duty free on your way out and pick up a bottle of Campari cheap.
If you smoke cigars, the Cubans at the duty free are generally considered to be real.
