Collision mesh should have Armature modifier added as well

That's what I refered as reference pose. One with un-applied transforms

Multiple $model statements are not allowed, use $body instead

And you mentioned (one) exported SMD while there are multiple smds in qc...

Exported mesh is built against reference pose of armature and mesh objects, not that you see in blender.

In qc there might be $definebone lines that re-define reference pose.

No, they won;t be kept for model, but you can make animations, and all constrainted bones motions will be re recorded and exported as well.

I did never import .qc file as it was always buggy and did import many unnecessary information.

Does the model exist actually?

There wan an error during compile. Put compile log here.

Vertex Groups are just kept. There is a trick, you can just change armature of model to Valve's one and re-name respective vertex groups to Valve's armature bones, though the result might be not so perfect.

in blender, enable Weight Link Cull Threshold. Re-export model.