I'm not sure but it very likely dd. The brakes are activated by 12V electricity so if there is no battery in it, the brakes can't be actuated that way. The break-away would be a switch that is held open by a plastic plug that has a cable to attach to the tow vehicle and pulling out that plug closes a switch to send power from the trailer's battery to the electric brake solenoids.
It probably does have a 30A, 120V power cord and plug. It is normal to have to replace them on occasion due to wear and overheating. Since the RV is not 33 years old it would be very unusual for it to have the original power plug in good condition. It should look like this image or close to it.
View attachment 175376
It is entirely possible that the extension cord was used in a defective outlet that had arcing or some other problem, or that it was used to the limit for too long and so overheated and was damaged. An outlet in poor condition can cause major damage to the plug that is connected to it. You may want to take a picture and post it here. To check out the wiring you need a reasonably good, multimeter and then start with checking for shorts to ground and continuity of the circuits. You can also use it to check to be sure that all of the circuit breakers are in good condition. Keep in mind that most of the appliances and the lighting are all 12V and get power either from a battery or from a device called a converter that converts 120V-ac power into 12V-dc power to supply the appliances and recharge the RV battery.
It sounds like your RV had a built-in generator at one time that has been removed. They are fairly common in motorized RVs. This is another situation where a picture or two would probably help us to know for sure.
If the existing water heater is original, it probably uses only propane to heat the water and has 12V power for the controls and ignition of the propane. It might have an electric heating element as well and that would use 120V power if it does. It isn't a terribly difficult job for one who knows a little about RVs and it should not require any disassembly of the RV.