Does your cousin have Applecare? If so, Apple should fix whatever is wrong. If your cousin is having problems with Applecare, he/she can ask to speak to Apple Customer Relations and perhaps they can help sort out the difficulty with the replacement optical drive.
Fans should idle at ~2000 RPM. If you have one at 10 RPM, it very likely needs to be replaced, and probably the other one as well since it may well be compromised by having been doing the work of two. The high temperatures are likely due to inadequate fan operation and hence inadequate cooling. Hopefully the high temperatures have not led to any damage--the computer is supposed to shut down first--but the fan problem is something that absolutely needs to be fixed.
Probably the best thing to do is to find the original grey system discs and run the extended version of the Apple Hardware Test, which will be on one of them.
Alternatively, make an appointment at the genius bar if you have access to an Apple Store and have them run their hardware test. Running a hardware test should be able to give you a good idea of all the things that could be wrong with the machine. That will give you the best information about what is wrong and what it might take to fix it.
If it's just the optical drive and the fans, the Mac could well be worth repairing. If something really expensive is wrong, like the logic board, then maybe not. You would have to weigh the costs of the repair against putting the same money towards a new Mac.
Also, Apple offers a depot repair for around $300-$350 if your Mac qualifies for it. This means no drops or spills or any other type of accidental damage. This might be an option to consider looking into if you get a high repair estimate.