What is the big panel that is raised behind the canopy on the F35, that looks like a giant air brake, but I noticed was raised during a traditional, long runway, take off?
The big 'lid' covers a large fan which blows downwards in the 'B' version of the F-35 when it wants to hover, or perform a short take off. The take off you saw in he program was not a conventional take off - it was a simulation of a short take off from a carrier, hence some downwards thrust required, thence the doors all open. There are similar barn door openings on the underside of the jet. The fan is positioned just behind the cockpit - the other 'non jump jet' versions of the F-35 (the A and C) have an extra fuel tank here which almost doubles the fuel capacity and therefore range. The fan is coupled to the main engine via a shaft and clutch, but the clutch only engages when the fan doors open. The exhaust nozzle of the jet pipe also pivots downwards. Therefore, when in the hover the jet is balancing on two 'jets' of air, one cold from the fan behind the cockpit, and one (very) hot from the normal tail pipe.
Also they quoted a top speed of 1200 mph for the F35, but the old English Electric Lightening could at least do 1300 mph (apparently, but subject to it's "secret" classification). The fastest Mk24 Spitfire in level flight could do 454 mph. So is that quoted "1200 mph" for the new F35 Lightening purely for the public's ears and is in actual fact a classified far higher speed than that? For £100 million, in 2019, I would expect the F35 to go a lot faster than the old Lightening!
Top speed is largely irrelevant - it's not an interceptor that needs to race off after a target. It's a (stealth) strike fighter, and in RAF service will (has!) replace(d) the Harrier, Jaguar and Tornado. There is no real need for speed - you ain't gonna outrun a Mach5 SA300 or SA400 if it manages to get a lock on you so better to sneak in at moderate speed (less infra red signature) and take out the SA300/400 at extreme range. All your weapons and fuel are internal so your radar signature is as low as it could be. Racing in at medium/high level at Mach 2 with external fuel and weapons isn't going to end well for you.
The other problem with going fast is you burn a LOT of fuel in re-heat (afterburner) and that drastically reduces time in the air and range. Many fighter pilots have flown fighters in the past capable of Mach2+, but if you talk to them they very very rarely went supersonic, and almost never went Mach2+. Indeed, many fighters advertised as Mach2+ can only do it 'clean' meaning they aren't able to do it whilst carrying external weapons or fuel/drop tanks. When loaded out with a realistic payload they'll struggle to even go supersonic, particularly at low level. Whereas the F-35 carries all its fuel and weapons internally, so can do the 'book' speed regardless of what it's carrying.
For the RAF/RN, the F-35B will be a mud mover chugging around at medium level tying to avoid detection by anyone and striking it's targets. The Typhoon will be the fancy Dan fighter making sure no enemy fighters get close enough to cause damage. AWACS will hopefully be several hundred miles away detecting hostiles and relaying their position to F35 and Typhoon so they can either decide to engage or employ tactics to avoid.