Ken-- thanks very much for the additional information and offer to help.
For anyone else who's reading this thread, one point of clarification and correction. In the posts above, I stupidly forgot that the restore itself would be creating new forums (duh!). Thus, I was wrong above that it's creating IDs for forums which don't exist. The new forums are created during the restore and then subscriptions are linked to them in mdl_forum_subscriptions. However, there's still the issue of the repeated duplicate subscriptions. (The same thing is happening in multiple courses and on my dev Moodle.)
I'm thinking this patch will fix the issue: https://tracker.moodle.org/browse/MDL-60669 I installed 3.4 a few days before the patch was integrated, so I'll need to update. I'll post here if that fixes it. Still frustrating not to know what's causing this, though.