I was able to get the rings from the "new" used differential from market place installed into my painted differential without any issues.

Now I've already sealed the differential, used my old housing case and pinion ring, but installed the ring gear, probably the same thing that you refer yo as crown gear, from the differential I purchased on market place.

Unfortunately I don't know how to measure backlash or gear tooth pattern, never took apart a differential before.

What I'm thinking about now is whether the old pinion gear and old ring gear will match together properly.
When I turn the end where the driveshaft bolts, seems to be turning ok on the sides