It's reasonably safe in Russia. Regarding driving style, Moscow is a just a huge city (perhaps more pop than in whole Switzerland). Perhaps people in New York drive similarly aggressive.

The question is whether you plan just to visit Moscow / St.Peterburg or kind of go backpacking across the country (the latter requires serious preparation).

BTW, OMON is not a regular police, it's a sort of police special forces to suppress riots etc. (literally OMON is "squad of militia for special purposes").