Very low oil can cause lack of lubrication which leads to all sorts of nasty things. The limited slip clutches may wear, the gears may go, the bearings may disintegrate. All very bad.
Old oil kind of depends. Do you mean drity grungy worn out oil or just clean oil that has been in there a long time.
Dirty grungy etc, has the same problems maybe worse, as low oil.
Old oil kind of depends on how old. A few years is probably OK as long as the diff has been kept out of standing water and lots of dirt. I would image diff fluid that has been around 20 30 years might "gel up" which would be a problem, but I would not expect this for most diffs kept in a clean environment without a lot of heat.
In any case, to be safe I would change the oil. If in doubt change it, diff fluid is cheap.
If you have a limited slip differential and do not change the oil often enough the XL-3 friction modifier additive can wear out. Then the clutch pack goes bad and has to be replaced. You can tell this is begining to happen when you get shuddering on turns. This happened to me, thats how I know.
2002 Explorer XLT, 4x4, 4.0L SOHC
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