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 ...
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.
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 ...
While researching information for this question, I stumbled upon the answer but thought I would post anyway to share my findings.
The Bricklink entry for the 9700 Technic Control Center set has a secondary picture which is the front cover of the instruction booklet.
The 5 Electric, Plug Holder 4.5V/12V elements can be seen in use as holders for the ...
It's more or less obvious that tracks themselves are compatible, although blue rails won't go on dark gray sleepers, which isn't really a problem. You may want to use more sleepers under blue tracks to stabilise it, although the 12v inserts help. Mixing blue and grey track and inserts shouldn't be a problem, should you wish to do so.
Motors are also ...
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 ...
I had one of those, and afaik it was just used for the lights. I had like 2 light poles in my set and they where used for that. lights is about the only thing that could take AC current instead of DC in lego. Those weren't LED's at the time. I just can't find any pics online from the ones I have(had...).
Edit: Found it, it was powering this light brick:...
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. ...
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