Can you copy public ssh id to authorized keys upon login without ssh-copy-id?
Besides using ssh-copy-id client side or manually pasting your key into authorized keys server side, is there a way once you have already logged into ssh with your password to automatically add your key? Presumably your public key has already been sent when you tried to login, so is there a way to tell the sshd daemon to add the key to the authorized keys?
NO.
When you log in with a password your private ssh key is not used (ie you can log in with password authentication even if you don't have a private key). Thus there is no public key for the server use.