@paolaelectra Sorry about the marked solution; someone who isn't me might have acted a bit prematurely. I can unmark it if you want (you can do it as well), and of course I would like to keep helping if I can, and if you have more issues. Threads don't get closed without reason, certainly not because someone says an issue is solved.
The buzzing might be as simple as the fan not quite being seated correctly. If the buzzing does only happen when the fan is running, not any other time, that would be my first guess: that it's wobbling a tiny bit as it rotates. Or it could be slightly brushing up against something else, although that seems less likely. You can open up the case again and spin it yourself, to see if you notice anything. If the fan wobbles a bit, it might need to be reseated, or the fastener (whatever it uses) tightened a bit.
You could also take the computer to a shop if you'd like a more thorough examination. In that case, it would be good to ask if the shop can reapply thermal paste to the processor, or more specifically to the contact between the processor and its cooler. Given the age of your computer, the existing thermal paste has probably dried out somewhat, and a new application could help with cooling. But if the cost would be high, it might not be worth the likely small improvement.
Using an external fan isn't going to help as much if the room where you're playing is hot, but it can still help to some extent. The important thing is that your computer isn't taking in the same hot air the fan expels, so even a little bit of circulation can make a difference. You won't see an improvement in the processor temperature though, because it's already thermal throttling. Up to a point, better cooling will just mean it can work harder before it throttles, not that it will never get to the temperature where it throttles. It's the same as before you cleaned the fans: the max temperature hasn't changed, the difference is how much the processor can do while staying below that temperature.
The most important thing though is whether you can play the way you want. If you're still able to do that, then the thermal throttling doesn't matter.
