I pretty sure I'm correct about DME feeding the FMS - how else does it know the track miles to run? The INS needs updating. It can come from GNSS or DME. I'm guessing the FMS uses DME as default.
You are correct except that the first thing you do when conducting a RNAV approach is turn off the DME update.
The FMC builds a profile from the end of the runway with fixes at certain points based on a 3 degree slope. It's all done in the computer with no external reference.
I always do a 3x check from the end of the runway anyway to ensure we don't have a Air NZ scenario again. (Type 'A Free Lesson' into YouTube if you don't know what I'm talking about)