For your first question TM should backup in straight round robin to each disk. If that is not working then perhaps the external drive needs testing. Or is actually not available. When a disk is unavailable TM will continue using the internal disk.
If the internal disk is one and the same as your OS disk then you have no backup. A disk failure will cause loss of everything.
A second question: I notice tat the backups don't, in general, include backups for the OS. Is there a way to 'force' a backup that includes the OS?
No, it is not possible nor required. Time Machine makes one backup of the whole disk. It then makes a complex incremental backup which is shown at the start of the backup.
Please read the basic info from Pondini on how TM works.
http://pondini.org/TM/Home.html
The first two articles. How TM works and how it is different to a clone.
My recommendation is you buy a backup software .. eg Carbon Copy Cloner and make a bootable clone on external drive. That can then give you an extremely simple method to test the quality of a backup.. hold option key during the computer boot and select the clone as the boot drive. This will give you a 100% test of the quality of backup in a few minutes.
