LO,
I sincerely apologise if I've not been sufficiently clear over the way UK requirements influenced the design.
As I think I've already said, the UK's involvement at the early stages (CALF, JAST) was pretty limited. At that time, early studies were informed by the UK's NST 6466, which certainly did call for a replacement STOVL aircraft capable of operating from an Invincible class ship.
However, as nab points out, it soon became clear that getting a larger (over 50,000 lb) aircraft on a CVS was never going to be much of a starter. The requirements for the technical demonstration phase (the X-planes) were very generic, and didn't go into much detail. However, the USMC and made it plain that the new aircraft would be required to fit into roughly the same footprint as an AV-8B. That started setting STOVL size.
The only formal requirements for the F-35 were set out in the JORD, somewhere around 1998/1999/2000. You will remember (but others may not) that this was the culmination of around 5 years' requirements development, and a key piece of that process was making sure that it didn't contain any items that inadvertently lead the design into a bad place - your example of the TSR-2's massive landing gear, (driven by a strange requirement for soft field ops) is an excellent one. Anything before the JORD was general and indicative - the JORD was the document that nailed down exactly what the customers wanted.
By the time the JORD was being nailed down, the UK had already decided to go for CVF, but they were miles away from a defined ship. As a result, the UK decided to put in a minimal set of ship compatibility requirements that would not drive the aircraft anywhere strange. Thats why the requirements were set out in the way I explained in a previous post - 'compatibility with CVS flight deck and hangar layouts'. The folding wingtips for the F-35B were a scheme (not even a design) offered by LM for discussion with the UK - but they were turned down pretty smartly, as the UK had deliberately omitted reference to CVS lifts. Like I said, the UK MoD were terrified of applying additional national requirements that might affect weight and cost. They were by then aiming to design the ship around the aircraft.
On the gun, you're on the money - LM weren't paying very much attention to weight in around 2001, which was a major error for a powered lift aircraft programme. The through life cost estimates they used were skewed to show the Gatling in a good light, the Mauser in a bad light. But, as I've said, water under the bridge now.
Hope this helps,
Best regards as ever to those making the trials work,
Engines