if the diff is working correctly is relatively easy to find out, no?
lock the left wheel (sprocket and such),
jack up the right rear wheel and turn it.
Observe the diff housing, if it turns twice as fast as the wheel it's working,
if the wheel doesn't turn at all, it's welded shut,
If it's anything in between (usually with noise) it's broken.
Unfortunately it's not totally impossible,
that some idiot replaced a straight axle with a diff and figured,
he could have the sprocket and single disc brake as they were before.
'sid