Night (with stars) skybox
I changed the UnlitGeneric's to LightmappedGeneric as lpfreaky said, still getting the same visual glitch. I'll try to find an alternate way to simulate night environment.
Thanks for the suggestions so far!
"UnlitGeneric" is what the standard Portal 2 skyboxes use. I don't think "LightmappedGeneric" makes much sense, since the 2D skybox doesn't have a lightmap applied.
Remember to also remove the "ignorez" line if there is one, otherwise the skybox will fill the whole screen whenever it's visible, hiding the rest of the map.
Other Portal 2 maps: Medusa Glare
Portal 1 maps: Try Anything Twice | Manic Mechanic
Space Skybox worked for me in a simple test map (the way I sent it to you);
Just noticed you might want to copy say stars01dn.vmt and rename/replace stars01bk.vmt ...
The bk texture file is the one with the planet earth image.. the rest are all just generic starry skies...
PS The test box room was a simple square box, 2 walls & floor textured with anim_background and 2walls and ceiling textured with tools_skybox. Map properties skybox name set to stars01. All textures (vmt & vmf) in materials/skybox...
No luck so far... my purpose is just to convey as good an impression of night-time as possible.
Earlier I had made some holes in the ceiling and inserted the tools/skybox into them. When that didn't work, I inserted brushes into the holes and textured them pitch black. But I also put an env_fog_controller, checked its Master flag, set Enable Fog to Yes, min distance 1, max distance 8, fog density 1, but I STILL don't see any fog!
This time I've attached the vmf of the map (which is structurally very simple)... I don't get why the stars skybox is showing the visual glitch, despite following RustyDios' instructions.
In your .vmf I found 2 major mistakes related to skybox:
1. You should apply texture tools/toolsskybox to all faces you want your sky to be seen.
2. In map properties you should set SkyBox Texture Name parameter to your skybox name (stars01 if you use RustyDios' texture). You have sky_black_nofog (btw that nofog part doesn't let you have fog in your sky).
In env_fog_controller set propper values for fog start and fog end (let's say 1 and 2500, not yours 0.000025 and 0.0025 or so 
) and choose a good color (somewhat bluish or darker). And experiment with these parameters.
A 3d skybox might be a better fit here. Like the one in the super8 demo.
PortalStories.com
"Oh, in case you got covered in that repulsion gel, here's some advice the lab boys gave me: DO NOT get covered in the repulsion gel."