When I attach geo to the agent, current geo normals are replaced with newly generated ones and don't react to any changes in the source normals. Is it possible to prohibit the regeneration of normals on an agent?

This is a limitation with Hydra currently - it doesn't perform skinning on the normals and instead flags them as needing to be recomputed after the deformation
A workaround is using the Bake Skinning LOP which does support deforming normals, with the downside of the skinning being baked instead of happening at render time

It's not look like a Hydra problem, because I made the same test with Unreal Engine. For instance I export skinned character from UE to usd, and back, and normals was rebuilded.
On the other side Bake Skinning LOP is not a good choice,
because it bake into transforms matrix and point positions, which interpolated linearly in usd, and you can get squash & stretch effect between animation keys, and bad motionblur as a result.