Found a better way to transfer .map to .obj after reading the tip somewhere on quake3world.com. Instead of q3map2 one loads the map into Q4Radiant (of all things) and then simply select everything and export selected as .obj.
This is the q3map2 version – as we saw before, it looks like a shattered mirror. The geometry of the version exported from Quake 4 (upper image) is very clean, and a single “tris to quads” in Blender gives you a workable mesh.
This will save me a huge amount of time.
In theory, this also means that you can easily map in Radiant and export to Unity. You just need Quake 4.
Currently still looking at engines, not decided yet. It can be done in both Unity and UE4 though, I’m sure.