score:11
There is no law against dirty cars in Russia.
However, your license plate must be readable (from 20 meters) and dirt shouldn't be falling off your car in chunks because there is a fine for unreadable license plates and "damaging the roads" or "hindering traffic by making the tarmac dirty".
In my own experience (that's 15 years out of date, tho) it's advisable to wash your car before driving into a city, especially Moscow. Better to give no reason for the police to stop you at all.
However, regular roads are nowhere as dirty as the extreme pictures you've posted, so in general your car will have just a thin layer of dirt and you'll be okay.
EDIT 20 Feb. 2019:
Back in Russia this week. About 1/3 of all vehicles in and around Moscow have illegible license plates due to dirty roads (still not nearly as dirty as on OT's pictures, tho!). According to the local driver, the law is not really enforced because the fine for illegible plates is only RUB 500 (EUR 7) - and it's 50% off if you pay promptly.