1. The old records can not be changed but from now on you can very well mention your name in stead of her father. Even passport does not require name off father even if parents are not divorced.
2. No, this can not be done unless he adopts the child with the consent of your ex husband.
3. You can start mentioning only your name in all her records as mentioning name of father is no more compulsory even in school admissions . But you can not mention name of your second husband as the father of your child without a valid adoption.
Payment of alimony has nothing to do in this matter.