The red car that you mentioned is Mail Van (7820).
There are a number of other trains in the picture including:
7730 - Goods train (the locomotive doesn't seem to be visible)
7710 - Push-Along Passenger Steam Train (just the cars)
7750 - Steam Engine with Tender (pulling the cars from 7710)
7814 - Crane Wagon
7760 - Diesel Shunter Locomotive
7720 - ...
You can absolutely run RC and PF trains on 12V track. You can leave out the center conducting rail since it won't be delivering any power to the train, unless you are planning to run both 12V and remote trains.
The main LEGO train sets have all used the same rail gauge, which is the main thing that matters.
If you'd like to use both your 12V track and the ...
All LEGO train tracks have had the same gauge, so newer trains will fit old tracks, including 12V/4.5V. Since the wheels themselves are not exactly the same, you may run into some small issues at points, but it should be OK most of the time - and using RC track wouldn't fully prevent the issue anyway. (If anything, I suspect trains woudl actually derail more ...
The 12v train motor uses 2-pole 3-slot technology.
Because of thinner, but longer wire which results in a greater resistance and will create higher voltage, but lower current the current output isn't enough to drive on-train lights.
Since the resistance of on-train lights is constant this will drag down the generated voltage as well.
As you can see in the picture below the friction has to do with driving the worm wheels via regular gears while pushing the unit by hand and yes, I guess the gear teeth on the wheel axles could be damaged because of the gear ratio.
A user on YouTube posted the following video that tests a lot of the various train motors in Lego history for their pulling strength.
The battery model train from the 1960's destroyed the competition.
Pulling Power of LEGO Trains 1966 till 2009 (Systems: 4.5V 12V 9V IR)
The newer engines do not have rubber traction to grip, so they have a tendency to slip ...
While I can't speak of the handling of these parts, the possibility of receiving official replacement parts from TLG is surely zero. That leaves us only the second-hand market, which fortunately offers genuine LEGO replacement pieces, sometimes even in brand new condition. Note though, that even though the brittleness problem might stem from lots of play or ...
I think it's for test purposes if the fixed and/or regulated 12VDC outputs don't work anymore.
This output is fixed 13VAC so it might be directly connected to the secondary coil of the transformer before it is rectified by a diode bridge.
With this output it is very easy to test if the transformer is broken or the electronics without opening the unit.
It's difficult to answer definitively without a photo or video of the issue, but I'm not aware of a way to easily get replacement parts for this.
I'm sure you're already aware that you can buy a replacement motor on Bricklink, but that's going to set you back at least $100.
Internally, this motor drives a pair of gears on the axles using metal worm gears. ...
I don't have this set, so I can't provide a fully authoritative answer, but 9V is likely to be safe for these motors. The hub that drives them is definitely 9V, as it connects with the rest of the 9V system and runs on 6 AA batteries:
TLG typically overspecs their motors somewhat, so you could probably also get away with running the motor using 12 volts, ...
According to Bricklink, Set # 7730 has this motor :
bb12va Electric, Train Motor 12V Modern Type I with 2 round contact holes
For disassembly and repair you can look at this large post with pictures on Eurobricks by Andromeda
The answer is yes.
Using the 13v~ (alternate) output of the grey line transformer you can command the blue line switches.
This is not in the manual of the new transformer (it is a kind of hidden door to the past) and the blue line transformer switches output was 16v~, but I've tried it and it works.