We stayed at a few convents and monastaries when traveling around Italy, and for the most part, they are really inexpensive - but the bargain ones are really bare bones. There is a really decent, inexpensive place in Venice though you can't make reservations and just have to show up the day you want to stay. There is a convent we used to live across from in the center of Bologna that takes women travelers though I have never inquired there (perhaps the next time Kei comes to visit).
These photos, from the place we stayed in Noto, Sicily, had no window screens in the sweltering summer and I was swatting mosquitoes all night - and the road leading out of the town where this convent sits was traveled by night truckers and their motors were incredibly loud. In addition, when we checked in, the mother superior interviewed us before accepting our request (to make sure we were married and because we were to sleep in the same section as the nuns, that we weren't potentially dangerous, which I completely understand. Check her out below- she was a tough cookie.).
But the nuns were so sweet and curious, and brought us fresh fruit and lemonade to our room, though they were not happy when we were 15 minutes late on the 10 pm curfew. We wouldn't chose again places that have a curfew - or lack screens if traveling in the warm months. I think air conditioning, unless you are staying at a high end place, is not assumable to say the least. When we left, they invited us to come back the next summer with the new baby. Ha! Catholics.