I'm having a similar issue with 1

The game launches into a black screen which occasionally flicks to the game for a brief second. The only way I can view it is to alt-tab which reveals the game running in a 800x600 window.

I've tried altering the settings in the DOSBox configurator but nothing seems to make a difference.

I had a similar issue with Unreal Tournament on Steam which I fixed by editing the .ini file. Will that be the case here?

I tried tinkering with the Game Graphic Options myself, I was able to set the resolution, but it only works when not in full screen somehow, as soon as you set it to fullscreen, it only uses the "scaling engine" with has x1 x2 x3 options, that's it.

Not sure if this is the issue, but you can switch between full-screen and windowed mode in any DOSBox game by pressing ALT+Enter

Bit late to this conversation, but anyways.

2 things for you to try.

First of all use notepad or your favorite text editor to edit the dosboxLOL1.conf or dosboxLOL2.conf (depending on the game) You'll see a setting under [sdl] called output=overlay. Try another option such as opengl (preferred) or ddraw or even surface. (e.g. output=opengl)
Also in the same section is fullresolution=original. Try changing that to your desktop resolution. (e.g.fullresolution=1920x1080) you can also try fullresolution=desktop, but not sure if that works in the bundled GOG DOSBox. Oh and "fullscreen=true" should be set to start the game in fullscreen and not windowed mode.

Secondly, check to see what your video card scaling options is set to in Windows. For NVidia cards, go into the NVidia control panel and choose "Adjust desktop size and position" If scaling is set to "no scaling" then that's likely the issue. For aspect correct scaling choose "Aspect Ratio" and for stretched full screen choose "Full Screen". "Perform scaling on" should probably be set to "GPU"
Not sure how scaling is set for ATI cards, but it's likely similar.

Thax for your tips. This is what worked for me (see screenshot) playing in Win11 and nVidia GPU. After that LOL1 runs full screen, in aspect ratio (a nice square) and it's pleasure to play it again. BTW - I cannot run with scaling on GPU, I get blank screen and then game closes, but with scaling on Display works like a charm!