As DireWolfM14 has suggested, I think Tails is what you need. It's the safest option, given the fact that you don't have a hardware wallet or a separate computer dedicated for signing transactions offline. Make sure to verify the signature of the OS img, and recover your wallet on the pre-installed Electrum.
Can you put private keys directly into Coinbase or Coinbase Wallet?
I highly doubt. Usually, they give you an address to deposit your bitcoin.