Can I check if the cooling system of my Macbook Pro 15" Early 2013 works as intended?
a faulty charging cable can cause all kinds of problems. That could have started your issue and it may have grown from there as it is what is sending power to all of your macbook, it is not limited to what it can break. 

one question I have is what are the circumstances for the use of the laptop. meaning are you using the laptop on a desk? on your lap? 

you can always get a laptop stand that has fans built in, and will help cool the system, and keep it from overheating due to being on your lap, or a blanket, something like that. 

the temps you are seeing are pretty high, here's an article with some people discussing the temps in their MacBook of that year.

as far as finding parts, I would suggest talking with the best MacBook experts on the planet at Rossman Group as I'm sure they can find the part, and will be sure to diagnose any problems and where they stem from. I spend quite a bit of time on his youtube channel watching him repair boards, and other issues, he is one of the true original MacBook and apple product repair geniuses. Plus he's pretty funny with his rants.