Florida recognizes all lawful marriages no matter what country the marriage took place, but I don't know about Russia. First off, there is a rebuttable presumption under Florida law that a child born to Wife during the marriage is fathered by the Husband. The parties can either enter into an Agreed order otherwise or request paternity testing. Anytime a paternity action is filed or a divorce action in which there are minor children, Florida laws mandate child support be paid by the party who has the least time sharing, formerly known as the non-primary residential parent. This is calculated as per statutes.
In a divorce action with minor children, the parties will either agree to a Parenting Plan or the Judge will order one. Your questions about custody all boil down to the Parenting Plan.The Florida statutes define a Parenting Plan as "a document created to govern the relationship between the parents relating to decisions that must be made regarding the minor child and must contain a time-sharing schedule for the parents and child. The issues concerning the minor child may include, but are not limited to, the child's education, health care, and physical, social, and emotional well-being. In creating the plan, all circumstances between the parents, including their historic relationship, domestic violence, and other factors must be taken into consideration."
As for him asking her for money to leave the country, that sounds like blackmail, which is criminal in nature. If the parents can't decide and agree to child issues, then they have to fight for what they want in court and prove what is IN THE BEST INTEREST OF THE CHILD. Is moving to Russia in the child's best interest? [Rhetorical Question] I do not know Russian laws so I can not answer any specific question about what would happen there. My guess is if Alimony is ordered in one country and the wife tries to get more in another country, the Husband would prove what he is already paying and the amount would be offset. I do not know if the Wife would be entitled to receive either of the types of Alimony in this case because that depends on many factors including length of marriage and wife's work capabilites along with her need and husband's ability to pay, etc.
No one can promise what will happen in Court, you just fight for your client's rights and try to prove to the court what is in the best interests of the child. Please note terminating parental rights here is not an easy task unless the father is beyond a complete deadbeat.
I strongly recommend that both Wife and Husband hire an attorney for the divorce proceeding.