I have to drop suppor for 3.3 LTS, due to incompatibility of geometry nodes for curvature and radius shaders.
From now on focusing on 3.4+, namely 3.6 LTS.

They do not, Diagnostic shaders use standard Blender Material system.
To see them, use Material preview or Rendering shading mode.

Due to potentionally long shader compilation times, the decision was made to limit these shaders with current implementation.
There is future possibility to apply sections / slicing for existing materials on the model, depending on the number of requests.

Yes.

Currently there is no simple way to change color of 1 curve in Blender.
Wire color from active Blender Theme is revealed in Addon UI to change it globally.
To revert to default, just press backspace while howering over the color button.

Curvature Graph modifier is currently limited, by using vertex groups for isolation of vertices/edges.
Either properly plug-in correct vertex group into GN Mod input, or make sure no faces are applied to vertex group.
2 neighboring edge loops will result in face selection.
To apply CGraph to two neighboring edges, just apply separate CGraph for each.

Note: This limitation could be overcome by using custom edge domain attribute, instead of vertex group. 
However due to a bug in Bevel operator/modifier (faulty distribution of attributes on created geometry) this is not currently possible.

In Eevee this is the default normals behaviour.
To fix this, just activate "High Quality Normals" in Addon UI, or Eevee Settings.
There is some performance drop associated.

To fix this, please turn off "GPU Subdivision" in
Edit / Preferences / Viewport / Subdivision.
GPU Subdivision inteferes with subd data handeld to Geometry Nodes Modifier.

Curvature data are calculated via Geometry nodes modifier. This allows realtime editing and flexibility, but at a performance cost. This node setup was heavily optimized for speed during development, but still can be laggy on denser objects. To go around that, lower subdivision levels, or select subset of faces to perform diagnose on.