Cleaning electric heaters
The recommended way is better than using another way to clean this type of heater.
Or you can try both of them, suck it with vacuum first then blow the rest.
The soot part of black could be some part of your heater burned up, either the plastic part or just another dirt.
The other way to clean up is removing your heater completely and try to disassemble it and after that, you can clean each part perfectly. (careful with the cable line, make sure it is disconnected both Live and Neutral)
The water type of cleaning process is not so good with the electrical device because it tends to leave the water in it and make it worse.
Edit answer from edited question:

For coil and rheostat : i always use rubbing alcohol or IPA (isopropyl alcohol) other than that i use contact cleaner because it is made for electrical.
The fan : same as coil and rheostat but i won't touch the inside of fan (the coil of fan) because I'm afraid if there is any residue left in the motor it could burn the motor.

Let it dry after cleaning with alcohol or contact cleaner.