Having fallen into the hands of the Organs of the State a couple of times in Russia I always carried my passport (if it hadn't been sequestered by the Hotel) but overall it 's not too bad - I really like Moscow. Out of town you just have to play it by ear - we used to carry a large supply of duty free cigarettes (non-smokers all) left carelessly, and very visibly, on the back seat of the car for example.
I can assure you that no cigarettes were ever used as a bribe at a roadblock - but IF the officers were helpful then of course it was only good manners to offer them a cigarette or 20 AFTER we were cleared. And word gets around, smoothing one's path through the day to day irritations of life ....................................