Why does CHANGE ENGINE OIL light come on and stay on for about 15 seconds before going out?

What is your question?

I know there is fairly fresh oil in it, so this is more of a nuisance than anything. LOW FUEL also comes on shortly after fuel tank is filled, may be a related electrical issue? Sometimes the fuel issue happens while driving, and then the gauge pops back to "full". The dealership guy said, as long as you know the oil is ok and fuel tank is full, don't worry about it! ??

What makes this problem better or worse?

Doesn't matter; always happens when engine is cold or hot/warm.

How long has your 2001 Chevrolet Silverado 2500 HD had this problem?

About a year, or little less. Mentioned it to dealership but they didn't deal with it.