Sounds like a heat problem, often a big issue with laptops. Your first description started focusing me on the video subsystem, but when you indicated it couldn't find any bootable devices, I think maybe the problem is system wide. I am unsure how comfortable you are about opening up a laptop, as it is much more involved and requires a much finer touch than opening a PC case, as it can get scratched and/or cracked.

When I had unusual problems with a laptop at my old job, I did the following. PROCEED WITH CAUTION AND DO SO AT YOUR OWN RISK!

Open up the access point to the processor if possible, if not; open the case to expose the processor. Carefully remove the heat sink (I think they are all different, so you have to figure it out for yourself) and see if the thermal compound hasn't dried out or whatever. If it has, it is probably the problem. Regardless of whether it is dry or not, you need to clean it off and replace it with a new thermal compound. I used Artic Silver, replaced the heat sink, and solved the problem. Understand that if you remove the heat sink, and it looks OK, you will still need to clean off both the processor and heat sink and put new thermal compound back on the heat sink.

When I succeeded with the above, I thought I might as well do the second laptop we had, as it was the same model and age, so I figured I would take care of it before it was a problem. When I was removing the heat sink, the processor came out of the ZIF socket with the heat sink, even though I hadn't lessened the tension. I was able to separate the two and replace the thermal compound with the Artic Silver, and the system ran fine, but it underscores the possibility of problems, so again, I suggest you be careful. You may be able to get a tech company (local computer store that offers repair service) to do this for you at a nominal cost if you aren't comfortable with doing so yourself.