Can a faucet leak when water isn't running?

Assuming that you mean when the faucet is not turned on, the answer is yes. A faucet can leak internally & externally any time water pressure is present. Before repairing/replacing the faucet, check for leaks at the input connections to the faucet.

Yes, several ways too. Do you want to talk about repairing this faucet, if so more details would help.