As of now, my core catalog and associated backups target my internal drive. These files are sync'd with Dropbox and further backed up to an external hard drive.
Am I making a mistake targetting my internal drive for backups? If so, should I target the external physical drive for backups instead? If I do this, the external must be plugged in to successfully backup, correct? What happens if Lr tries to backup the catalog and the external is not plugged in?
You are doing fine, because eventually you are backing up to an external drive. Backups should be made to an external drive, so you can use these if anything happens to your internal drive. Lightroom could also make its catalog backups directly to the external drive. If that drive is not connected when Lightroom wants to make a catalog backup, it will warn you about that and offer you to temporarily use the internal drive. That is better than no backup at all, of course, and you are backing up these anyway.