Does anyone else suffer from both seats being wobbly, the plastic mount seems very flimsy.
A 'fix for this would be appreciated

Can't say I 'suffer' thereby but the seats do have a bit of movement if you shake them. Never heard of someone breaking one though. I actually never much gave it a thought. 

Are any of the rubber stoppers missing from the underside of the seat?

Rubber pads, I was thinking about thicker pads, and yes there are pads there but thin.