Can you have your backpack when you go in museums and churches!
It varies.
If you can't, generally there's a stashing facility.
A shoulder bag or crossbody bag is not usually a problem. Many churches and museums don"t allow backpacks - imagine toppling a Bernini sculpture with a quick turn!
Never at the Galleria Borghese - they will keep it and give you a plastic bag with your valuables.
A small one OK at most churches, take it off and carry it.
Most museums allow a small bag, most have lockers (put in a refundable €1 coin and remember to collect it when you empty the locker), don't wear the backpack because it is too dangerous.
By the way - are you Holley or John or Karen?
Although my cross-body bag stays close to me and has been acceptable at just about any museum, I still relish the chance to check anything heavy and just move my day-wallet to my pocket.
Never had to store it in a church but it varies in museums. Lockers will be provided if it's required in most places but you may also run into where they will just store it in a cubby hole behind the counter or another room which I never liked but never ran into problems with. Just make sure you take any valuables or passports with you. Some museums that allowed you to carry it would want it either carried in front of you or hand carried as oppose to over the shoulder which never made much sense to me cause as far as I can remember of those places it wasn't crowded with displays and you couldn't get close enough to one anyways to accidently hit anything. You may find erratic enforcement of the policies also. I had an emloyee at the Doges Palace tell me I needed to store my smallish backpack after letting a group of 25/30 teenagers walk past us most of who were carrying much larger, some camping size, backpacks.