From my experience, that does not sound right.
Dumb question: Does the differential have any oil in it? Are you sure?
If it does there is something seriously wrong to generate that kind of heat in such a short drive.
Is the exhaust system routed properly? Do you have the repair records to track exactly what the claimed repairs are?
'03 E320 Wagon-Sold
'95 E320 Wagon-Went to Ex
'93 190E 2.6-Wrecked
'91 300E-Went to Ex
'65 911 Coupe (#302580)