While you could export your private key from Bitcoin Core to Electrum, it's safer to use sweep feature which available on Electrum or simply send Bitcoin from Bitcoin Core to address generated by Electrum wallet.
That's correct but that's also depend on what he's trying to do. If he wants to prove the ownership of an address (and not the funds), sweeping won't be of too much help because that would only move the funds from his Bitcoin core to one of his Electrum addresses.