Belsirk: could be something very dark in your machine, if after all theses step s is still present
FowderSoap: My machine is fine. Did a total virus scan, reinstalled keyboard drivers, did a Windows Update...
Redownloaded the GOG installer (same bug applies to Fallout 2 as well), installed it to the Desktop, tried the suggested patches, ran it in 95/98/XP compatibility mode, renamed ddraw.dll...
It has nothing to do with the computer. I remember playing this from a physical disk on my parents' computer years ago, and it was buggy back then. The game was just poorly optimized from the start, and getting it to work has only gotten harder with each successive OS generation.
Belsirk: the best will be contact Gog support.
FowderSoap: GOG discontinued support since they no longer sell it.
Do not take personally when I say could be something "dark in your PC". The PC and laptops are a wide world, and there can be different behavior from different hardware with the software and O.S.. Maybe, something is wrong with the updated driver for your keyboard that Alienware don't care, maybe they don't care to fix it because the problem appears with apps from 20 years ago or older, or there are not reports to them.
By example, my same laptop with Linux Mint, Windows 8 and Win 8.1 had different behavior with Firefox 34-36, with Win8 Firefox just unresponsive and become unable to be killed. This was with the same hardware and software but different S.O. and drivers (as the drivers are made for the S.O. to know how to operate the hardware there are big difference between Linux - Windows drivers , and sometimes from version to version such as Win8 - Win8.1.
As result, some times the bugs one got from apps become very unique as are not related how the app/game operates internally but how the communication with the hardware is handled. With Fallout is more tricky, because truly they overwrite all the interruptions calls related to the keyboard (and maybe with other hardware).
As I said before, your problem is not a well-known bug for this game, do not means you are lying or you are the culprit by having that specific model or something like that, only that is going to be hard that the community can helps you more than the previous suggestions. Maybe with a Virtual machine or in another device you can play the game. Or maybe using the community patches could give the solution.
I only can say that I played without problem Fallout 1 and 2 with Win 7 and Win 8, using 64 bits processors with little of the community mods (just the fixer and wide-screen mode).
Though is very bad that gog truly don't give technical support for those games anymore.