@DivieOwl @CyrusTheLoser @sunblond
Problem solved, I found the culprit CC. I've informed the creator about it, it can be fixed, right?
So I deleted the CC and when I went back into my game to edit some of my sims who now had no pants on (since I deleted them) I noticed that one of my sims who had been continuosly falling through the floor when walking was also wearing the pants. Seems like that problem's gone as well.
Here she is walking down the stairs, with a plate in her hand:

Bottom line is, it seems my problems were all related to this one specific CC.