Thanks for your reply. I thougt about that doing it in godot by creating multiple collision shapes (and/or bodies), if this is possible. But this was just a thought.
If I got you right, I should make a second mesh as collision mesh, but with less details. Will this work with the suffixes mentioned in the documentation (Import Hints)? So I have to name the second mesh "CollisionMesh-col" for example and it will be imported as the collision mesh?
I wonder how such a fps controller will work on terrain or other bumpy ground.
Another thing I found was the suggestion to increase the physics frame rate, but this was not helpful. It is a little bit better, but not what I want. I think it just increases the rate of collision checks, so some miscalculations couldn't come up (at earlier tests I popped through the floor often).
And yet another thing is, I found the project setting "Smooth Trimesh Collision" was set to false (default). After changing this it gets much better (no sliding along some edges anymore). There are some issues here and there, but it seems to be usable for.
But anyway, if I found out how to, I will follow your suggestion and try to do it with seperate collision meshes even if it is more work (many, many models to rework).