Skybox Issue

What's wrong with it? I can't see any problems.

That's what I was going to say. It looks like it's working fine.

afaik you CAN see displacements/props/func_details through skybox. there isn't anything "wrong" with it.

If you can 'see' one part of a displacement, the whole thing is rendered

Yeah, I can't really tell what the issue is with the skybox from that shot either.

World details will render in front of skybox IF they are in a visible leaf.

Probably what happens is from that view you can see into the leaf, so they are there. But when you move you move into a new leaf that can't see the leaf they are in, thus the disappearing act.

Instead of having a square corner maybe have a 45 degree angle. Or expand the height/width/whatever of those areas so the leaf issue doesn't happen.

The skybox DOESN'T have to be really tight to the gameplay area. You can have world area the player just can't get to due to walls/fences/player clip. Have some extra room behind it, then have skybox.

Hard to tell also, but it looks like your skybox might be too low, make sure you have enough room for a demo to get at least a 'one sticky' jump in without hitting his head.
Demos can go so high with 2 stickies I don't generally plan around that, just too hard to optmizie your map AND let demos have full jump height. But it sucks to hit your head when only using one sticky.

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And I think something is wrong with your hud, I've never seen that before

You're trying to see the red bit?

If a face is entirely blocked by another block, or is not seen on the "inside" of the map, then it is removed entirely from the bsp.

Are you referring to how you can see that other CP through the skybox? Yeah, the skybox doesn't hide stuff that lies beyond its brushes. Options include:
-Adding cliffs or some building to hide it.
-Using hint brushes or shortening your map's render distance so it doesn't show up until you are close enough.
-Area portals, somehow?

Wait I see what's going on here.

Shouldn't that be intended? If not, you should just optimise your map better, or put something in (like a cliff or building) between that area and the CP.