I know how to create a paper wallet and send bitcoin to it from Coinbase. If I want to spend the paper wallet bitcoin do I just give the merchant my public key? How does the network know I hold the private key from just that private key info? If I want to send the bitcoin to someone do I have to download the entire bitcoin protocol to do that? Is there a smaller version I can use to just send bitcoin? Thanks for your help.
If you want to spend bitcoins from that paper wallet then you have to import the private keys to another wallet and send them.
You could use either online wallet like blockchain.info (don't recommend it) or some offline wallets like bitcoin core/multibit