I am in Litigation as Plaintiff in Federal District Court of New York. Defendant filed a Motion to Dismiss. In my Opposition to motion to dismiss, can I request Discovery for Admission and Production? Do I need the courts permission?

Barbara, as far as I know you can start discovery once you are at issue. Discovery is your right and from what I've seen in the course you need to start discovery from the beginning. That had been my mistake before. You need to gather intelligence and info from the beginning.
Be sure to check the rules of the court especially the local rules and follow them. In some courts, you have to file interrogatories and Admissons with the court and in others you can just send them to the opposite side, but you need to find out and the local Rules are sometimes hidden traps. This is not legal advice (my disclaimer--i could be just dreaming or sleep writing.)