Can I recover a TDE certificate by restoring the MASTER database? (We are, fortunately, not currently in this situation, just planning ahead to see what our options would be if it ever occurred.)

For a database encrypted with Transparent Date Encryption (TDE), a copy of the database backup is unrecoverable unless you have a backup of the certificate used to encrypt it.

What if you don't have that? Are there any additional options?

In the event of a total server failure, would restoring a backup of the MASTER database on new hardware also restore the certificate?
Since TDE relies on a certificate stored in master (which is used to encrypt the database encryption key), then this would work only work if you could restore the master database to another server in such a way that the certificate could be decrypted.

This is the TDE encryption hierarchy:


Service master key (protected by Windows; tied to the service account credentials, and a machine key)
Database master key (in this case, the one for the master database)
Certificate
TDE encryption key


The first three items are stored in the master database, and can all be backed up. The fourth is stored (encrypted by the certificate from #3) in the header of the encrypted database.

So in a failure scenario, you would have to restore enough of the encryption hierarchy to allow you to read the TDE key. SQL Server creates the service master key at installation; thus while restoring the master database to a different instance will also restore items 2 and 3, the necessary key(s) to decrypt them will not be present. Result: unreadable data.

The two best options are to either restore the certificate (#3) from a backup (a good option if master cannot be restored for whatever reason), or restore your master database and its master key (#2) from a backup. Restoring the master key may be a better option if you have a lot of certificates/keys protected by this key, and need to make them all accessible at once. This comes with the same precautions normally associated with restoring the master database (collations, logins, database names and file paths, etc.)

Generally, I'd only recommend restoring master in a recovery scenario. For a migration/scale-out scenario (such as using Availability Groups/mirroring with a TDE-encrypted database), it's better to backup/restore the certificate (#3) so that it's encrypted using the master keys unique to each instance it's moving to. You will need to include the private key with the certificate backup.

In any case, you're going to have to make key/certificate backups, so guard them well, and store them in redundant, secure locations. Simply having a backup of master will not get you out of a TDE disaster; you're going to need a backup of at least one key or certificate.