Can you check your balance on line with visa/mastercard or only if your bank supports it?
Yes, go to Bank of America and open a Visa Checking account. You can only spend what's in your account and you can use it like a Visa at any place that takes it, has the same 16 digit numbers as a Visa. Plus, you can set a spending limit on it each day. And yes, you can go online 24/7 and see your account, what transactions you did, what checks have cleared, etc.
BTW, I never see the teller, I get Direct Deposit to the account from my employer and deposit any addittional checks into the ATM. Then order checks online.
I do use Paypal and that's a different story, I have that linked to a Wells Fargo Savings account where I keep near 0 balance, just in case Paypal tries to charge me back. I also keep my Paypal almost 0, as well.