Good evening chaps,
after spending another two hours in the garage I finally solved the locked seat problem. I had a friend help me by pulling upwards on the seat really hard (the plastic is quite flexible so nothing broke) so I could get to the locking mechanism. I ended up using a wire cloth hanger similar to Barbel Mick's pic to manipulate the position of the tube and then pliers to pull on the tube which finally worked.
I also solved the mystery of the faulty locking mechanism. Turns out some absolute donkey installed the other end of the wire the wrong way which robbed the mechanism of about 1 cm of wire which explains why the teeth didn't come out far enough. In the picture you can see the correct way it is supposed to be; I drew the way the wire was installed before in red.
Full res images and working animated GIF here:
https://imgur.com/a/8y7bTDaAbout the wobble: Now that the lock is working properly, the seat is a lot less wobbly overall. In fact, I think I will try removing more spacer rectangles because I'd like my feet to be able to touch the ground a little more firmly. Has anyone got experience with doing this?