Hi @kszwaba,
Thanks for the detailed report!
The X server responsible for your ThinLinc session generally won’t have access to the GPU, even if the user has the correct permissions. So for example, if you want to run accelerated OpenGL applications in a ThinLinc session, you’d need to use something like VirtualGL. This is why glxinfo is showing a fallback to software rendering.
ThinLinc shouldn’t affect things outside of a ThinLinc session, though. So if you’re unable to get sensible output from vainfo at the local console for example, then you would need to resolve this first.
I’m not familiar with VA-API at all, so the following is pure guesswork 
but if/when you have vainfo working at the local console, you might like to try unsetting the DISPLAY environment variable before running vainfo within the ThinLinc session. This will (hopefully) prevent it from trying to use ThinLinc’s X server for GPU access, and get it to use the console one instead. So something like:
DISPLAY= vainfo
or
DISPLAY=:0.0 vainfo
Note that you shouldn’t change the DISPLAY variable globally, for example by exporting it, since ThinLinc relies on this being set correctly.
Let us know how you get on.