In the continued fashion where STW can answer any question ever...
The wheelchairs are specially made for the sports, usually made of steel, but I've seen a few titanium ones. Very expensive, a titanium one with decent wheels will be around £3000-£4000 I think.
You are correct, spinergy wheels and components are seen now and then.
Due to the fact they don't have to take many impacts and not a huge need for stiffness, you occasionally get plastic spokes instead of metal.
I can only talk for indoor tyres here, but most of the time I see what are pretty much road/track/wheelchair tyres. I'm struggling to think of any names right now, but from companies you'd see at any Sunday club run.
We usually run around 120psi or so, never heard of anyone running tubs or tubeless or anything like that. The tyres get very unevenly worn due to the angle of the wheels (imagine if you constantly turned left or right on a bike), and need flipping over after a while.
The chairs have quite a low centre of gravity and are very stable, but people do still fall out occasionally. I've been umpiring a game where the rear wheel detached itself which caused the player to bang their head on the floor as they toppled over backwards. Thankfully she was ok, although shook up.
I can't speak for much on the tennis side of things (I'm a wheelchair badminton coach), but if you've got any other questions, or are interested in getting involved at all, I'm happy to point you in the right sort of direction. Great photos by the way.