I have bought two games: Sim City 3000 and Vampire: The Masquerade – Bloodlines.

Both of them don't work on my computer. As soon as I launch them, they minimize and stay minimized. If I try and restore them I get a second of sound, a black screen, and then they minimize again.

I requested a refund of Sim City 3000 but then I got the same issue with Vampire.

I don't get this issue with any Steam games, so it's not a general issue with my computer and seems to only happen with GOG games. I want to support DRM-free but obviously I'm not going to if none of the games actually work. I own Vampire on steam and have played through it without issue with the fan patch applied.

Does anybody have any advice or should I just refund Vampire as well? I also bought Alpha Protocol but I imagine that will have the same issue.

Both games start and play flawless in fullscreen for me (Win10/64bit, single monitor setup).
What are your specs?

Have you checked the subforums, or pcgamingwiki already?

https://www.gog.com/forum/simcity_series?search=sim%20city%203000
or
https://www.gog.com/forum/simcity_series?search=simcity%203000

and

https://www.gog.com/forum/vampire_the_masquerade_series#1711317700

respectively:

https://www.pcgamingwiki.com/wiki/SimCity_3000

www_.pcgamingwiki.com/wiki/Vampire:_The_Masquerade_-_Bloodlines (remove the_ after www) Let's not resort to hyperbole...you own 802 games here.
If these two really don't work (and they should work, mind you!), then that's ~0,25% of all your GOG games.

Pretty far from: "none of the games actually work".

Windows 10 64-bit, too. I suspect it's because I have a 4k monitor, but it might also be the GPU I've upgraded too since last trying an old GOG game.

I have checked various GOG game forums. There are lots of threads by people reporting what sound like the same issue, and no real solutions, just silly workarounds like changing your resolution to an extremely low one like 800x600 and 16bit colour every time you start the game. You're right and it was also an irrelevant detail, but to be fair you're assuming that most/all of the others don't have the same issue. I've only even tried a handful of them (12 according to my stats, and only 4 of them have more than an hour played). Most of them were free either on GOG or Twitch, or bought in sales just because. I have even more on Steam and use that platform a lot more than GOG, and still have only installed a small percentage. I have more money than time. As an older gamer I'm sure you can appreciate what's that like, and your time is more precious as a result. Having to waste it is frustrating.

SC3K: Yeah, I don't think exclusive fullscreen works on it anymore. Try -w (or -W) as an execution switch. (See your operator's manual on how to do that.)

VTM:BL: Don't know.

Use this app in order to check whether a program - and if so, which one - is causing the trouble on your machine.

This isn't the only complaint I've seen recently about 4K monitors having trouble with retro games. I wonder if someone out there knows or is working on a universal solution?

You should provide more info. Do you tryied old game like simcity 3000 from steam? It is big difference runing relatively new game and very old game. In most cases Microsoft is at fault or GPU driver.So it is better ask for support on nvidia,microsoft support center.