Can I check if a Mirror database was synchronized when the principal died?
You can check the mirroring_state_desc in sys.database_mirroring.  It will tell you if the databases are in sync, this does depend on if you're running in High Safety or High Performance mode.  In High Performance mode as long as the mirroring process is running it will show state of SYNCHRONIZED while the actual state of the database can be behind as the log records are still being applied.

In this disaster type scenario, I believe the mirroring would be broken and it would be best to know the various ways to get a tail of the log backup combined with the normal course of database and transaction log backups to restore to the point of failure.

Tables 5-7 in this TechNet article cover the various scenarios against the statues reported.