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I'm not gonna guess about the humming noise. My 83 300D doesnt hum and It could be several things. most likely your guess about the differential is right. Dont worry about the oil, it will be jet black within 50 miles of changing it. It gets black, but if you use good diesel oil it can easily last 4000 miles between changes.
The "cooler" on the side of the radiator is the oil cooler. Since it cools the oil, your comment about the oil film on it sounds like it may have a very slight leak on one of the fittings. not to worry. little oil leaks are very common, and usually painless to fix.
The humming noise would be my concern at this point. may be worth checking the diff fluid prior to purchase for abnormal metal shavings or leakage.
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Matt
2003 Jetta Wagon TDI 145,000MI
1999.5 Jetta TDI 287,000MI
2001 Suzuki SV650S 26,000MI
2008 Yamaha Vino 125 11,000MI
http://www.crazymatt.org
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