Burning smell

That warning light is for oil pressure, not level. Have the pressure checked with a mechanical gauge. If it really is low, as the light indicates, suspect worn engine bearings. If it is not low, suspect a defective sending unit. Those aren't real expensive.

Nope. A common cause of the smell is oil leaking from the oil pressure sending unit. (When you smell hot oil, it can be leaking from anywhere). A sending unit that reports incorrect pressure / turns on the oil warning light when it shouldn't is often accompanied by leaking past an internal seal. That would be the place to start looking. The sending unit will usually be near the oil filter.

Valve cover gaskets often leak oil onto the hot exhaust manifold. You'll smell that. There's lots of other causes of smells like electrical problems, sticking brakes, and accessory drive belts.