Backup:
Plex must be stopped when you make the backup. If it is not, then you risk database corruption. At a minimum, backup the database files, since those define your libraries, etc. Since you’ve custom artwork, you’ll also want to backup the full Plex Data Folder.
What backup application to use is up to you. One my servers runs on a Synology NAS, and I use their backup application (HyperBackup) to backup the Plex Data Folder. My other server runs on Linux. On that one I manually make backups once every couple of weeks. There are backup apps for Linux, I’ve just never bothered to use one.
Moving the Plex Data Folder
Yes, it can be moved to another drive. If moving to an external drive, you must take care that the drive is not disconnected when Plex is running. If that happens it will corrupt the database.
Exactly how to move the data folder depends on the operating system.