Is it possible to stay logged into Bitwarden after closing down the browser? (in my case Opera)
Yes, this is possible. However, it is insecure as the vault will be written to computer’s disk, which could be targeted by malware (theoretically).
To do so, open Bitwarden browser plugin and choose Settings, then check “Security” column. You can choose option “Never”.
Where is the vault written to disk? Is it still encrypted once written to disk?
He meant the encryption key will be stored to disk. The encryption key that unlocks the vault needs to be unencrypted in order to unlock the vault. Who ever has the encryption key can unlock the vault.