I agree, Ken: ATMs are always the best bet. We've used them at the airport and at banks with no problem but check with your financial institution to make sure they'll work before you go. At the same time, raise your maximum withdrawal amount, if needed/desired, and let them know when and where you'll be traveling so they can put a note on file. Same with credit cards.
We take photocopies - front and back - of our cards, airline reservations and front page of passports. Husband and I each have an ATM and credit card and so back up each other. They are only carried (Pacsafe or small pouch pinned under our waistbands) on days we think we may need them: otherwise they stay in the hotel safe with the passports. According to a different post, you've ordered a Pacsafe, right?
Yes, we've always had to produce our passports for copying when checking into Italian hotels. We also had to produce them when buying our Friends of the Uffizi passes in Florence so expect that you may be taking it out now and again. Nothing to worry about.
Yes, different ATMS will dispense maximums of varying amounts so if the first transaction is rejected, try a lower dollar amount.
We don't travel with phones (yet) as we just don't want to mess with them. Should we need to make an emergency call (i.e. a lost card) we'll bite the bullet and just pay the cost of doing so from the hotel. So far, so good. Yep, we're still very much Paper Map People but it works for us, and I use a mobile device for checking mail from free wifi spots - which more and more hotel rooms have.
No, not all businesses will take a credit card, and especially not for small purchases so always have cash on you. It won't apply as you're with a tour but we've had to pay cash at some small hotels we've booked. Anyway, our pouches are easy for us to access (but not so for someone else) and we'll often carry a small amount in a trouser pocket that wouldn't ruin the day if it was lifted. Again, so far, so good.
But most importantly, while it's wise to take the usual precautions the bigger issue is not to think that you have to stress about it all day, every day, every time you leave the hotel room. The small minority has has a bad experience - and many of those made that all too easy - and the majority have not. We keep the same wits about us abroad as we do in any large American city so it's really just second nature.