Uh... oh.... wow, I'm surprised it's still even remotely working at this stage. But really, get your fan fixed first before you even plan on doing anything else.
Well I ended up convincing my dear mother in renting me a new laptop, which is godlike compared to the one in my first post. So now, the problem has taken a backseat-concern, but I'm still going to try everything in this thread, thanks everyone.
the main causes of BSODs are numerous, heat being the most likely here, but memory errors are another one, there are diagnostic tools from Microsoft and Who-crashed can analyse the crash dumps.