It seems like the majority of the older games have mouse issues with the newer computers based on all the games I've downloaded. The mouse on almost every game is sluggish and stuttery. It also seems like the movement left to right is not bad but up and down is like molasses. Is there a generic fix for these old games for slow mouse responsiveness? Does it have to do with the screen resolution? It's sad because I've downloaded lots of games and would like to play them but it's too frustrating when the mouse moves like molasses.

hmm, I have a lot of games from GOG, but never encountered such a problem.
Did you had a look if there are newer drivers for your mouse?

Which games are you talking about specifically? Do then run via DOSBox, or are these early Windows titles?

I've never had such problems myself as well, but I'm still using XP... the Win/DOS question is also important.

It sometimes happen for DOS based games. If you have mouse with adjustable DPI, change it to the lowest possible option.

Just throwing a wild guess here: specifically, when playing FPSes, moving diagonally up-left has no issues at all, but moving diagonally down-right is where the frustration begins, as if the mouselook response just refuses to comply. Is that the issue? :3

(because that's the problem I'm having too, and very sadly, I haven't been able to find a solution for it at all)

In fallout, there was some dll-file which I removed and it worked after that fine, it was really sluggish mouse before. Baldurs gates mouse flickered, that was because forcing something propably.
I was freaking suprised that "alone in the dark" doesn't let config your keys.. Well, not so suprised. Good thing that it uses arrow-keys, still, propably a good thing to read the manual in future.
Old games, weird problems. New games, mostly compatible but usually a crap game.
Usually forums answer a lot. Still, dos-box could be better. Should be abit like scummVM or something.
Mostly I have trouble with sounds or v-sync.

Acceleration in logitech's driver system causes these problems, I know that for a fact. If you have acceleration on try turning it off.