Firstly, if the cover is loose, you are going to get excessive vibration as the motor turns to read to the disc. This could also be combined with it struggling to read, as they will often spin a lot faster when they are struggling with the laser. Buy a selection box of screws or try to find some replacements, alternatively, as a temporary measure, you could tightly tape round the loose DVD case and see if its actually working and worth the fix.
You could be looking at dried thermal paste under the heatsinks or failing capacitors. As the other comments say, the amount of time between turning on and turning itself off could point closer to the problem if shared.
This is indeed caused by the clock capacitor leaking onto the motherboard traces and these will need repairing or jumping using wire to fix. The CPU caps are all going on these too, but these don't cause any damage, just eventually fail so the unit won't boot up. The model of the board looks like a 1.2-1.4 so you can remove the clock capacitor completely or replace it. You'll need a set of CPU caps to replace the others below the CPU heatsink. I service these in the UK if needed and stock all parts. Also have friends in the US that can help.
