Is foreign country court decision regarding alimony, child support, and custody of children be bounding in U.S.?

The short answer to your question is 'yes', but watch out for all the negative ramifications of that action. I have seen numerous cases of foreigners getting divorces in their native countries then coming to the U.S. and not having their divorces count. One variation is one spouse running off to a country, filing for divorce and then sending a 'decree' to the stay-behind spouse in the U.S. Unless that divorce is domesticated here, it can be challenged successfully during a future divorce where a court may determine that the first marriage is still in effect and that the second one never took place because of the first. Stated another way, the second marriage is void if the foreign divorce is defective. Any foreign divorce can be challenged as defective if there is a difference in the law between that of the foreign country and that of the U.S. and in most situations, there is a difference (service of process, time to appear and answer, location of property and so on). Another variation is the one where the U.S. couple travel to their home country, get divorced there, but then return to the U.S. and one spouse alleges no divorce ever took place. This variation will involve U.S. property and children because of our absurd and excessive awards of child support here. I see these in Muslim family situations were the couple go to the middle east to divorce, but then the mother challenges the divorce because of the huge difference between the outcome of a Sharia divorce and an American divorce money-wise. It seems the safest foreign divorce takes place when a couple was married in a foreign country, one spouse arrives here and then travels another foreign country to get a divorce from the foreign spouse and returns to reside here. Many of these divorces took place after World War II when displaced persons realized their spouses were not killed yet the spouses married others and had more children. The safest divorce is the one where the foreign couple obtain their divorce in the foreign country and arrive here divorced. So I would caution you as to your divorce plans. Get your foreign divorce but domesticate it here afterwards. Be sure to watch out for procedural problems like residency requirements, service of process and the nature of the property distribution. Any glitch will serve as a basis for challenge and you'll end up getting two divorces. Good luck.

Best to do the divorce in NYS where you both live and have NYS courts retain jurisdiction over the parties going forward. A Dominican divorce may be enforced in NY if there was proper due process. Much easier to handle the divorce in NYS. Contact an experienced divorce lawyer to assist you.