In my experience, the photo policy varies from one Church to another. Some churches allow photos to be freely taken, some require a small fee and some don't allow photos at all. Those that charge a fee often provide a small coloured wrist band so their staff know who's allowed to take pictures.
As you've mentioned, flash is banned in just about all churches, galleries, museums, etc. This is especially important in art galleries where light from the flash can degrade valuable paintings. However there always seems to be a few inconsiderate morons that think the rules don't apply to them, and they flash their way through the building until they're caught and escorted out. They usually seem to be those with P&S cameras that haven't bothered to figured out how to disable the flash.
Tripods and Monopods are banned in many places, as are the frequently reviled selfie-sticks. I vaguely recall that Tripods might be banned in some street locations too, but I can't remember the details on that.
As mentioned previously, St. Peter's is very strict and those who break any of the rules are ejected from the building very quickly. That goes for the dress code as well. Both men & women should ensure that shoulders and knees are covered in churches, especially in St. Peter's. You can find more information on the dress codes in the RS Italy guidebook.
Last time I was in the Duomo in Milan, there were signs prohibiting photos, but as usual there were lots of people snapping away. I asked one of the Priests about the rule, and he just shrugged his shoulders and walked away. Some of the smaller churches just don't have enough staff to enforce the rules.
In the Basilica of St. Francis in Assisi, there were numerous signs all over the place prohibiting photos and videos during Mass, but again the photo morons made their presence known. The Priest halted the service and issued a stern reprimand, and even with my limited Italian, I understood what he said. The reprimand seemed to have the desired effect.
I haven't visited the Borghese Gallery (Rome) in a few years but last time I was there, there were NO cameras allowed so I had to check my DSLR. There were staff in each room so those trying to sneak a cellphone photo would likely be caught and escorted out.
I always check the signs at the entrance to each facility, and if necessary pay the fee to take photos. If no photos are allowed in a particular facility, I respect that even if others are snapping away.