I'm not convinced of a 4-car 166 - whenever there has been a spare middle car it has usually ended up in the middle of a 2-car 165. It also wouldn't escape the notice of photographers.
It's definitely happened - I've been on one. I can't remember the circumstances, but a couple of years ago there was a brief phase where two Turbo units (at different times) had an extra coach inserted. I can't remember the specifics and I probably wouldn't have it written down anywhere, but I could be sure it's been mentioned on the forums before.
A four car turbo formation could potentially work on the NDL but there would be plenty of spaces where it couldn't stop - eg Dorking Deepdene. I've always thought that the 6-car stop at Betchworth harks back to the days before Turbos. The previous fleets of course had shorter carriages.
Stations like Dorking Deepdene can accommodate 4 coaches of MkI stock (source: Sectional Appendix). At that particular location, the length is 87.8 between the tops of the ramps, so with some careful positioning the doors could probably all fit on the platform, unless you went for the much simpler option of locking out the doors on the rear coach at one or two stations.
Before the platform on the down main at Maidenhead was extended from 3-car length, closure of the down relief necessitated the locking out of a whole unit on a 2x2-165 working that stopped at Maidenhead. Given the short platforms on the NDL, I don't see why it would be different.
"Fire regs" (as they are known colloquially in the industry) will require a certain number of exits from each coach and each unit, thus you would not normally be permitted to run a formation with a unit where only one of its two coaches was accessible, as there would only be 4 available exits for passengers.
Turbos are not fitted with any form of SDO or local door operation, it's 'all or nothing'. The Bristol sets will be fitted with SDO but as part of newly installed guards control panel. Admittedly this would also work on the NDL which are guarded as far eastward as Redhill but would be an issue Redhill-Gatwick which is DOO. (Incidentally I've always wondered about this anomaly, particularly as the guard remains on the train for ticket duties!)
Some Redhill-Gatwick workings are DOO, not all, as far as I know. I seem to remember some of them, possibly some evening ones, are booked to run without any second member of crew at all. I'll have to ask a GWR guard next time I have a moment.
For Redhill-Gatwick, all platforms could easily accommodate any of the conceivable Turbo formations in use on the route, with the caveat that GWR services only call there in emergencies and therefore "once in a blue moon" etc. The shortest is Salfords which can hold 8 coaches of 377 stock. You can only dispatch from these stations with bodyside cameras anyway so a guard would need to be provided for that, without taking into account SDO.
I suspect that Network Rail feel they are in a position of power regarding proposed 3tph, which they do not like. It's being openly stated that anything that will potentially cause performance issues for Thameslink services heading towards the 'heroic' 24tph core frequency will be opposed, and have not exactly helped themselves by deciding to turn Redhill P1 into a south facing bay, and defining P0 as the primary platform for London-bound services, therefore all moves to/from Reigate will potentially conflict with the northbound timetable.
I do get the feeling that the TL Programme does need to settle down before this sort of improvement is tackled, but I wouldn't necessarily agree that the positioning of the turnback platform is going to be any worse than if it was placed anywhere else within the current boundary of the station. If Platform 0 had been the bay platform then trains to or from Gatwick would still have had to cross the entire throat at the south of Redhill in order to get to or from the BML.
However they are also in control of a solution to the level crossings issue (which is not limited to controlled crossings) and can therefore block introduction of the third train by tactically avoiding agreement with DFT and GWR on how the solutions will be funded.
I'm not sure that anybody currently has the authority or money to sort some of those crossings out. You're quite right that it's not limited to the controlled ones - indeed, I'd be more worried about risk and misuse at some of the AHBC ones which have no permanent monitoring solution - but I guess that the more rural and lightly-used you go, the less investment there would be to build bridges and underpasses. Take Buckland AHBC as an example: you could use a bridge, such as crosses a lightly-used footpath and farm track just to the West, and drop the road down a little further (taking into account drainage from the hills to the north) but who's going to stump up the cash when the whole success of a 3tph timetable could also be affected by numerous existing flaws, such as flat junctions etc.?
The Thameslink timetable has reserved 2 paths per hour to/from Gatwick for a regular interval NDL service but I will be surprised if the second path is utilised for a while.
So we have the paths; and the crews are there, indeed the trains could probably be there too. Oh dear!