Hmm. Strange crap happens with these computers, but this one is something I don't have prior experience with.
Given that the temporary solution is with the AC adapter, though, I suspect that the adapter might be part of the problem. Try replacing that, and see what might happen. Though with the power drain, it may be also the power supply unit of the computer itself. Check that as well; there's been quite a number of incidents in our shop wherein it was just the power supply that was acting up.
Might as well try checking the motherboard while at it. The BIOS ROM chip might be faulty, and it may only be a matter of time before the backup fails as well. There might also be a problem with the POST; it shouldn't take 5 minutes or so to set itself up.
When you say it 'turns off', do you mean 'turns off completely; no fans, no light, no nothing', or 'turns off all output, but still has light and fans on'? Because if it is the second one, the BIOS is indeed faulty, but if it's the first one, it may be the BIOS, the power supply, the adapter, or God knows what that is trying to take down your computer.
Godspeed, brother, hope this might help. Check back once you've got some news to share.
EDIT: Some more information.
The failure to boot up initially and the sudden miraculous boot up are probably both power errors. It's either the power is not getting as fast to the PC as it should be, or it fires it off in spikes. Either ways, that's going to fry your systems sooner or later.
I'm not sure about these guesses, but it's all I have from late-night searching technical forums (read: searching right now). Godspeed, brother.
There was once something here,
but it's now gone.