I believe that Russia is now accepting the Green Card
Green card isn't accepted, you can buy liability insurance at the border.
You'll have to fill out a customs declaration form which was only in Russian where we crossed. You might look for a translation online before you go. They print up a customs receipt thingy that you need to show when you leave with the little exit slip of paper.
If your name isn't on the registration, you need a translated and notarized power of attorney from the owner. If you don't have this, they turn you around and you get wait in line to enter Poland.
Translated copies of the registration and the technical document would be a good idea too, you might be able to get across the border, but it's a fine when you get pulled over. Make a copy of the IDP and your license and give that to the police when they pull you over. Don't let some ******* traffic cop try to hold it for ransom.
Beware of GAI. There are always little control points at the entrance to the cities. Remember to obey the speed limits, don't cross any solid lines no matter how faint they are, and keep your seat belt on until the cop asks you to come to his car to pay your "spot fine". Don't part with more than 1000 rubles unless you've run someone over. If they persist, ask to go to the station to pay your fine.
I don't know of anyone that rents cars to Russia. You would need full coverage insurance which is ridiculously expensive in Russia. It would be cheaper to buy a beater and drive it over.