When you think of thermal paste, think of a layer of jelly between 2 bread slices. The microprocessor is one bread slice (the actual wafer or the cap over it) and the metal plate of the heat sink is the other bread slice. The jelly is the thermal paste. The microprocessor sits flush against a heat sink (metal plate) to help pull heat off the processor and dissipate it via heat spreading fins and fans. But the surfaces of the microprocessor and the heat plate it's up against don't transfer enough heat just touching against each other and processors will burn up. The thermal paste (just a super thin layer) greatly increases heat transfer from the microprocessor to the thermal plate on the heat sink. The metal plates actually have little 'micropores' that make the surface less flat, so when the thermal paste is applied it fills in all the pores in the metal and conducts heat much more quickly to the heat sink.
The 'wrong kind' to me means just a lower quality, but really they all should work, if applied properly.
The exception to that might be a really thick paste (for example IC Diamond- one of my favorite thermal pastes) which is excellent but darn near impossible to work with. For a desktop processor, the weight of the cooler and the pressure of the clamps is enough to make the contact (this stuff is just super thick and doesn't really spread easily, but also is less likely to break down). For a laptop, it would be tough to use because the heat sink clamps and screws don't usually creat as much pressure, so it would take a lot of work to apply. And, if applied incorrectly, there are air gaps which don't transfer heat at all and completely defeat the point of using a thermal paste -- and fry the processor.
Having said that, I seriously doubt any manufacturer or tech would use it on a laptop without the person specifically demanding it - because it's expensive and a real pain to work with. T
The companies that make thermal pastes actually give detailed instructions on best application techniques for their product depending on how it spreads. The only real mistake is to not actually cover the whole processor, but it doesn't take much. Too much does nothing except overflow everywhere and create a mess and potentially short out leads around the processor and other components. (When you press the heat sink down on the processor, it squishes any amount that is 'too much' out the sides, so more is not better, just messier.)
Not replacing it, however, would cause your system to get really hot if the thermal paste has broken down over the years, which is normal. I'm really betting the tech just didn't replace it --> that's pretty common too. And to 'add more' may just give the easy-explained answer of actually replacing the paste.
A corrupt bios causing overheating seems less likely. possible. just less likely.
You might also download some fan control software and try increasing your fan speeds and see if that helps. It's not the permanent fix, but it might help in the interim.