Creating a backup of external Photos Library

Hi

Add the external drive to the time machine backup. (or better stated, remove it from the exclude list)

system preferences>time machine>options>click the external drive (currently excluded) that you want to include, and click "-" to remove it from the exclude list.

Hi

No the backup is most likely fine: a library cannot be opened directly from the time machine backup, or even on a drive used for a time machine backup. You must first restore the library to a compatible location (your system drive, or an external directly connected drive - no network location - formatted APFS or MacOS extended). Once in a compatible location, it will be possible to open the library.