First of all, the term "printing money" is figurative. Most money is electronic, and "printing money" consists of buying US Treasury notes and then electronically putting Federal Revenue Notes into an account. When you want to convert to paper, the bank calls up the Federal Revenue, the Fed subtracts the electronic money from the banks account, and shows up with a truck of paper money.

Yes. You can create paper that corresponds to a bitcoin wallet. All you need is to have an artifact with the private key and you have control over the wallet.

The problem that you run into with paper is that anyone can copy the paper, but you can encode the private key into a smart chip in a way that it is difficult to retrieve.