Unless you're going to pick it up theres no way to test it before purchase. Just buy one from eBay from a seller with over 500 feedback, 99% or more, and you should be fine.
Definitely sounds like a loose connection - usually in the hinges, in my experience. Screen cables get frayed and damaged over time. Swap it out for a new one :(
Check to make sure the router doesn't have MAC address blocking - it may be denying your laptop.
Its also worth checking there is not an IP address conflict - IE, two devices attempting to use the same IP.
CPUs do not fail very often. Motherboards do - especially graphical problems. I'd check and see if you can find a compatible CPU cheap, but theres a damn good chance theres a problem with the board - most likely the GPU.
I would not recommend opening a power supply. A lot of the components inside are seriously dangerous.
If you're dead set on opening it up the plastic might have just slipped - see if it will fit properly in place before removing it entirely?
That's harder to answer - could be any combination of hardware or software problems.
The best bet is to analyse the BSOD logs: http://answers.microsoft.com/en-us/windows/forum/windows_7-performance/how-to-read-dumpfiles-after-a-blue-screen-of-death/0d7522e3-4ed1-448e-bda6-66441f042c0c
I'm not too familiar with Playstation 3 partitioning, but perhaps you could make two partitions on the drive - the first partition as something the PS3 cannot read, and the second one as something the PS3 can read?
Your CPU temperature is extremely high.
AMD lists the max temperature for the CPU at 100c:
http://products.amd.com/pages/notebookapudetail.aspx?id=34&AspxAutoDetectCookieSupport=1
I'd seriously consider replacing the thermal paste and monitoring the temperatures.