Colnect is the one some folks from here are using for their banknote collections. Even though it does offer some swapping capabilities it is very basic and you basically negotiate the swap via private messages. Better than nothing though!
I think many governments frown upon websites showing modern notes. However, I use www.worldbanknotegallery.com. It is one of the more comprehensive sites I have found.
Yes I do. You can click in the appropriate box which notes you have and can download a list. You can even use it to list your doubles and swap notes with other members.
It is not 100% complete, nor is it flawless, but I like it. Have not found anything better.