1. Yes, if it disconnected and is unable to sync, you can still access the vault.
2. If you change the master password elsewhere, when your app is able to sync, it will log out automatically.

Ideally, since you want to write your master password down to keep it safe somewhere (because you can forget), you write the new password down before you change the password.

cc: @WtfEvenIsExistence@reddthat.com