I buy Euro in the US before leaving. I paid the same exchange rate for Euro I bought in the US and for Euro I bought in Italy. My bank also recommended a minimum amount of Euro I should buy, but I bought more than that.
It's nice being Euro-ready for snacks, a meal, caffe' (or a good stiff drink), and the Taxi. And it's nice having Euro ready for those lodgings that want payment at check-in.
Your bank will use your Account to buy Euro. If you want to convert cash, your bank may know of a good place that will accept cash.
In Italy, use your Debit card at a bank's ATM, if you have US dollars you want exchanged, go to Poste Italiane. They will ask for your Passport. I paid the same exchange rate at the bank's ATMs and at Poste Italiane.
Yes. There may be a transaction fee of (say) $5 imposed by a US bank on some accounts. Other accounts at the same bank may have No transaction fee. Ask your bank here about those fees.
It helps to withdraw as much as you can each time to reduce number of transactions, but some bank ATMs in Italy will dispense less than others for every 24 hours. So you might have to make more transactions, but like my bank said, if you need the Euro, don't fret about the fee.
Italy bank ATMs will read only checking accounts. So to transfer from savings to checking, use your iphone, and if your bank has an App, install that. In fact, your bank here may have a lot of useful information for you. Your bank needs to know your dates of travel.