The Axiom website calls this part the "Axiom Power Tower" here and lists its uses as:
Expanded environmental, life support, storage and payload capabilities
So it looks like the solar array is for increased energy production, and the "trunk" section will be mostly for storage.
Other than the Star Trek one, these images are of the controls of the JPL 25-foot space simulator. This was built in 1961 which is why it looks like it does. It is still in use, and Ingenuity, for instance, was tested in this. Here is a blog post from someone who visited it: you can see an image of part of the control panel there.
I'm afraid I don't know ...
What mission is this photo of Chris Hadfield associated with
Expedition 34/35, specifically the Soyuz TMA-07M launch crew.
and what is the location of the photo?
The Gagarin Cosmonaut Training Center in Star City, Russia
What is that ~30 cm mirror-like thing to their right (our left)?
It's a frame for the insignia of the training center.
Here's a ...