As a Russian citizen who's been to DME quite a few times and flew to and from the US I can say I've never seen, heard of or suspected any "illegal" briberies taking place at the airport.
There is a certain law that may prevent citizens from leaving the country if they have monetary debts to the state that exceed 10000 RUB, e.g. if they haven't paid taxes or fines in time. See the original text or a google translation of the law. In such case the citizen is indeed required to pay the necessary amount to become legally "clean". However, the payment is made to the bank account of an official entity (whose bank account number is well known and can be verified via the Internet) and cannot be considered bribery. Moreover, you can check your girlfriend's legal status online yourself at any time here and here.
The Russian customs have also been known to do some borderline ridiculous things like singling out travelers and arbitrarily applying some obscure legal clauses that not many people are aware of, such as calling your laptop or handheld console a cryptographic device without a permit, but all those checks were formally "legal" and the detainees were not extorted money from.
Good luck and please take it easy no matter how things go.