It’s a separate (from the webserver) database server.
No, in the panel on the mySQL databases page you will find a tool called phpMyAdmin, you can use that to access.
You should definitely keep a copy of your database offline. Using it offline might be a little bit different and more involved than you are visualizing. You could set up a local copy of WP and restore the database locally and read that way, or perhaps use a database tool locally to browse thru the tables in a manner much like you see in phpMyAdmin thru the panel.