How to safely resume a SQL Server Mirroring scheme that had a expired certificate?
Since Mirroring operates by shipping over a live transaction log to the secondary servers, you'll need to start a new mirroring session.

Your secondary database has been out of sync for a week and unless you've managed to maintain a transaction log on the primary that spans that entire time (hint: you haven't), you'll need to start with a fresh backup on the secondary to establish a mirror session again.

This is also faster than theoretically letting a week's worth of logs get re-applied to the secondary anyways.

As for the primary database's availability, yes it will be available during any sync (whether resuming after a pause or setting up a new mirror), because it's always sending transaction logs to the secondary but that does not interfere with it's own availability.