Plutonian, the situation doesn't sound too great. I would've expected overheating to be the problem, and it sounds like it isn't power given that battery-only didn't work, unless it's a power issue on the motherboard or something like that. Could try with only the cord and no battery, but I doubt that'll work.
Reseating the RAM might possibly help. Although I think that even without RAM, it would stay on for more than a second - you just wouldn't be able to do anything.
Reseating the CPU might also help. That isn't possible on all laptop - some manufacturers such as Apple use CPUs that can't be removed. But it's possible on others. A CPU issue might cause this sort of issue.
But other than that, it sounds like motherboard.
If you ever need to replace the power cord, look it up by model number on eBay. I got an exact copy of mine for $10-$15 shipped that way - probably from someone whose old laptop had died.
Another question: What are the advantages of using TIFF when scanning document-files? I tried looking it up and got some overly-technical camera terms that I didn't understand.
Thirdly, can anyone point me to a program similiar to WinDirStat, but without that huge headache-inducing weird graph thingy?
I use
JDiskReport. It has a few different views, none of which look like the WinDirStat image on Wikipedia, the free encyclopedia. There's probably at least one of them that will prove satisfactory.
TIFF is more space-efficient than BMP, but generally less so than PNG or even high-image-quality JPG. I guess given the choice between TIFF and compressing BMP yourself, TIFF might make sense, but I wouldn't use it over automatically-converted PNG or JPG. I use JPG when scanning myself, with the DPI setting high enough that I don't care about any marginal loss in quality due to the compression.
TIFF does seem to be more popular in scanning and on Macs, but I'm not sure why. I tease at least one of my friends about sending inefficiently large images via e-mails in TIFFs; he realizes it's less efficient and less compatible with web browsers than PNG/JPG, but apparently it's the default in some proggie

.
I thought "Run" was always run as an administrator. But you should be able to achieve the same effect via an administrators' command prompt and entering the command there.
I think general hating on the .NET framework and downvoting software only because of that is mainly by Linux and free/open source fanatics. It's by and large considered a first rate development platform, at least if your main target is Windows. To be sure, it has its flaws - it isn't as backwards compatible as Java, for instance, so if something uses .NET 1.1, potential users have to download that specific version, even if they already have 2.0, 3.0, or 3.5 (2.0 software will run with 3.5, however). But downvoting software solely because it uses .NET is silly.