There are some concerns about Lunar dust attaching to the space suits during the Apollo missions. So Suitports are being developed. On Earth, cars get pretty dirty too, but a car wash solves that. Would a similar washing method be feasible in vacuum or in thin Martian atmosphere, using some mixture of cleaning fluids or gasses? Or if I may add, cleaning by rubbing some powder or fluff nap to it?
You could, though gaseous nitrogen and a pressure hose are usually used for such purging of contaminants. Such purging systems were also installed on Discovery and Atlantis Space Shuttle Orbiters for the purpose of purging payloads.
Problem with water is that it's heavier for the task, it tends to freeze in and around the nozzle without constant source of heat and then block it, and, as a liquid, it also doesn't have the best cleaning properties due to its relatively low viscosity yet high surface tension, so it tends to clump and miss spots, so to say. It also gets things wet which can then freeze, or freezes in hard to reach places, expands as it does that and can cause damage due to it.
See Why was the Hubble Space Telescope purged with nitrogen gas? for description why nitrogen is a better for purging, and since you mention Apollo, also check How was dust-mitigation addressed during the Apollo program?
You do not have to use water you can A alcohol or antifreeze base solution to spray off moon Dust so it would not freeze This would have to be done in a small room not much bigger than Porta John this may have to be left on moon to you later time moon as far as the shield getting Scratched up moon dust you would have to have a multi layers shield even if nothing but something a little thicker than cling wrap