Laptop giving me a very hard time

Sorry for the non-descriptive topic title, but there appears to be more than just one issue (can't be really described within a couple of words).

It concerns an Acer Aspire 5043WLMi, with Windows XP (not sure what version was installed on it).

About half a year ago, it fell down and the screen cracked so rendered it pretty useless. After a long waiting time for the insurance papers etcetera, we all filled it in and eventually, after about 2 months, we received the money to repair it (as the laptop had already been sent to Acer, but awaiting an 'OK' to perform reparations). In the meantime, I figured, might as well add a tiny bit of money to the amount I received from the insurance and get a new one, which I did (now using the new one). If Acer does not receive an OK, they'll send it back, which happened.

I now figured to use the old laptop (with the broken screen) as a Media Center and keep it plugged in to a TV all the time. Before I tried installing Media Center software, I tried a clean Windows XP install, but it kept giving issues...

At first, when I inserted an installation disc and had it boot from disc, it would start the pre-setup phase; check and load driver files etc., after which it enters another pre-setup screen, where it checks all partitions and where you'll be able to create and delete partitions, but merely 2 seconds in the screen, the laptop shut down (no bluescreen, no heating issue). When I kept it running, it would continue in the XP install it had and just work fine (which proves it wasn't an heating issue?).

I tried several original XP discs, all gave the same result; the laptop would shut down (not closing properly, but just turn off) during the first 2 seconds of the second pre-setup for Windows.
One time, I left the XP that was installed running and checked the drives with Paragon Partition Manager and noticed 3 drives. The default Windows one (C), an extra one for data (D) and a hidden one (like 3GB or something).

Elsewhere, I was advised to try GParted Live CD and boot from it. So I tried that and removed all current partitions, created two new ones (with formatting), one C for Windows and applications and one D for all other stuff. Then I added the 'boot' flag on the C partition within the GUI and applied everything. Still no luck; the same shutting off occurs when trying to enter Windows Setup. If I kept it running now, it would display the following:

I could still access the PhoenixBIOS Setup Utitily by pressing F2 right after rebooting.

Due to circumstances, I wasn't able to do someting with the laptop for a while, but then wanted to get it working again, so started trying stuff (today).

I inserted a bootdisk (fdisk?) and let it boot from it to try and use the fdisk command, just few moments after it displayed the command line in DOS, it shutdown again. This was like 30 secs after i first started it again, in about a week. So definately not a heating issue. I tried checking the battery and/or cable. The LED was blinking faintly, when I unplugged it, the LED started burning bright again (no blinking), when I plug it back in it starts blinking faintly again. And now, the laptop won't turn on at all anymore.

Are their any solutions? Or is the laptop now basically completely broken (the battery perhaps?)

Apologies for this long story, but I wanted to make clear everything I did/tried was stated in order to provide as much info as I could. If there's anything you still need to know, let me know.

Thanks in advance!