FWIW, currently in Italy (Sardinia) since April 10.
Arrived via Singapore into Milan, various paperwork clutched tightly in hand (international vax certificate from Australia and European passenger locator). Walked up to the automated passport gate, straight through, then an automatic stamp from the nice immigration man. As we had only carry on, straight to and then through the unmanned green channel at customs and into Italy…….
We needed to show our vax certificate to enter Departures for the domestic flight to Cagliari and then again to board the plane. In each case, no problem, they just needed to count the shots, one, two three.
Since the, we’ve been asked to show our vax certificate exactly twice, both times in restaurants. One time, they were a bit flummoxed and called out to the boss that they had an Australian certificate which wouldn’t scan (as everyone here on AFF knows, ours is not compatible with the European standard) who immediately said no problem. The other time was in a country restaurant with very little English, so we were struggling to explain (our Italian is very limited!) so we just showed them the reverse on which we had printed the explanatory letter in Italian published by the Australian embassy that points to the Italian regulations specifying that the certificate meets the rules to be equivalent to the Italian super green pass. We had printed our certificates and the letter on the reverse and laminated it for just this situation.
As to masks, we are supposed to wear them on entering indoors anywhere. Museums etc are enforcing this strictly. Shops somewhat casually but it varies. Restaurants/bars fairly relaxed.
It may be different in other parts of Italy but both in and out of the cities, the above is the situation in Sardinia.