Do spacecraft engines suffer from carbon accumulation the way typical petrol/kerosene engines do? Just wondering whether the spacecraft engines/drives, or their booster rockets accumulate carbon the way car/truck engines do. What about ion/methane drives?
 A: Not typically. In fact, the opposite generally occurs. The high temperatures and velocities in the rocket motor tend to cause erosion (ablation) along the nozzle. There is considerable research into the ablation of the nozzles because it changes the shape and thus the thrust characteristics. See for example this paper, and a simple search will reveal many more.
It's also important to note that many spacecraft engines don't use carbon-based fuels. Solid rocket motors typically do, the binder material is usually a carbon-based material. But some liquid rocket engines are hydrogen and oxygen, so no carbon is involved. 
A: Actually, it is a factor in some systems. RP-1 is a grade of kerosene specifically intended for use as rocket propellant, engineered specifically for reduced breakdown and coking at high temperatures which can otherwise cause problems for regenerative cooling channels, turbopumps, etc.
Also see page 115-116 in The Rocket Company.
A: *

*If the LOX/Kerosene rockets would produce 
some soot, that would not stick to the combustion 
chamber, the velocities of the gases is just too high. 

*In fact all that  engines since the V2 up to Saturn V 
use an excess of oxygen or alcohol in general, but especially close 
to the walls. 
This excess component vapor layer protects the walls, which 
otherwise would melt away immediatly in a stoichiometric 
oxygen-kerosene flame. 

