1. If you can find your Thetford model you should be able to find a drawing to compare to those above. That will give your answer.
2. At least some Thetfords have a different hold-down bolt arrangement. A friend had that experience on his older Dutch Star. There are adapters available.
3. Old Biscuit is absolutely right about adding a shut-off valve while there!
It is very easy to swap out a toilet. Three years ago I removed the Dometic 210 and installed a 310 in less than hour! That included careful clean-up and being super-cautious because I'd never done one before. Three months ago I replaced it with the 320 in a similar amount of time.
I didn't put in the shut-off and have regretted it several times since. Reason: one should smear plumber's grease on the flap and seal periodically, so you want to have no water there while doing so.
BTW, the 310 is the best flushing RV toilet I've ever experienced, and I miss it. No need for a sprayer to remove skid marks. The elongated 320 is much more comfortable and I'll stick with it, but you definitely need the sprayer.
If Dometic can figure out how to get the same swirling action (as the 310) into a 320 upgrade, I'll swap toilets again!