Can I be on the mortgage but not the deed?

Sure you can... WHY you would even want to be is another question.

Yes you can, but I agree, no one would want to do that, since you would owe the money with no control of property

Sure, you are a co-signer. What is the benefit to you?

I'm not sure you can, especially these days. It's something you have to ask the lender. After all, they 'own' all but the front steps at first.