No.

No, they are unrelated.

You need to find out what's happening with your oil pressure pronto or it could spell a quick death for your engine.

Not necessarily. The thermostat measures coolant temperature, not oil temperature. Lack of oil will cause the engine to heat up, but it won't necessarily lead to overheating (hitting H) being measured via the coolant. You may or may not notice coolant temperature being slightly hotter due to the way the engine is cooled and lubricated.

Low oil pressure will cause noise due to metal on metal contact in the engine. The engine will stop running in an oil starvation situation because bearings, rings and seals fail quickly without the oil due to the friction. This does overheat the engine, but not in the same way as lack of coolant. Lack of coolant causes the heads, head gasket, engine block, etc. To fail. Lack of proper lubrication and oil cooling causes more localized things like the end cap bearings, crankshaft/crankcase bearings, wrist pins, timing chains, rings, cams, etc. to overheat. These things are directly lubricated and cooled by oil, not coolant. Coolant flows through the block and takes heat from some of these things through the cylinder walls and coolant jackets, not by direct contact. 

More modern vehicles like my 2017 Mustang have an oil temperature thermometer in its digital instruments as well as an oil pressure sensor. If you hear metal on metal sounds like tapping, knocking, scraping noises, etc. you definitely have an oil pressure problem. Get an oil pressure checking tool and manually check the pressure with the engine running. That will lead you in the right direction.