I'll give a slightly alternate opinion, and that is to not use any solvent inside the differential. You don't want anything that is a thinning agent left over when finished. And the labor to do a cleaning is not worth anything to the functionality of the differential. I would use my hands and a rag to clean out debris from the bottom area of the differential.

The burnt oil smell is a normal characteristic of used gear oil. It just has a strong smell.

If you use the Mobil 1 Synthetic gear oil, it has an effective cleaning agent that will take care of residue removal inside the differential. Let it run for a few thousand miles and then change the oil again.

The maintenance cycle for replacing Mercedes differential oil is supposed to be 30,000 miles a change. This is usually news to people who own American cars, because we are often taught the differential oil never needs to be changed. If you follow the MB maintenance cycle, the differential will likely last the life of the car.

Ken300D

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1982 300D at 351K miles
1984 300SD at 217K miles
1987 300D at 370K miles