Can I connect a mailbox with a different Gmail address than the account I sign in with?

No. For Google, the mailbox address is taken from the Google account you authorize, and the address field is locked to that account. If you try to re-authenticate an existing mailbox with a different email address, you get the error "You are trying to re-authenticate with a different email address. Please use the same email address." Microsoft is more flexible: you can sign in with one account and synchronise a shared mailbox that the account has access to.