1. To minimize the risks, I recommend that the father obtain a court order determining the place of residence of the child as with the father and such order shall provide for the mother to have parental access to the child. Such an agreement has full force and effect.
2. Article 32 of the Civil Procedure Code of the Russian Federation provides that the parties by agreement may stipulate to a jurisdiction where any issue would be tried. This Code applies to all matters, including custodial disputes.
3. You should also add Conflict of Laws provision: the parties agree that even were this matter brought before any court of the Russian Federation, the American Law will apply based on Articles 166 and 167 of the Family Code of the Russian Federation, and any dispute relating to enforcement of the agreement will be consistent with this provision.
4. The agreement must reference Articles 61-79 of the Family Code of the Russian Federation.
5. The agreement must also be signed in one day, by both parties simultaneously, and must be notarized by a single notary. It must be stated that each party is represented by his or her lawyer, and the notary must be a third party who is not related to either of the attorneys. There must be apostille and a notarized Russian translation of the agreement. To avoid misinterpretation of agreement (see art 431 of Russian Civil Code), I recommend to sign it in both languages at the same time.