Good questions raven4d. I try to include a lot of detail, but I forgot to touch on hardware. It's an Intel P4, when I look at "My Computer" it says 504MB Total Memory and 185MB caches, but the free memory is fluctuating around 7MB, so you may have a point. It says KDE is at 3.5.1 Level A.
The following is the output of df (to answer your hard disk space concerns):
Filesystem 1K-blocks Used Available Use% Mounted on
/dev/hda1 16835512 4218588 12616924 26% /
udev 258240 136 258104 1% /dev
/dev/hda3 25245368 519092 24726276 3% /home
/dev/hdb6 57976720 30709400 27267320 53% /archive3
/dev/hdb1 19550436 6251980 13298456 32% /archive2
/dev/hda2 114202588 56310304 57892284 50% /archive1
Someone I was talking to at a Christmas party suggested maybe the machine is trying to hibernate and only getting part-way there. That's another thing I can look into (I now have three: your memory/disk idea, my overnight backup idea and my friend's hibernate idea).