If custody was just awarded to me a month ago, can the mother file for custody. If she doesn't serve me do I have to go to court

Custody can be modified once every two years or when there has been a change in circumstances. I assume the mother originally had custody and custody was changed to award you primary physical custody, given the child is 16. If that is the case, the mother would have to show a change in circumstances to change custody again. Of course, she can file anything at any time, you would just be able to dismiss the action because there was no change in circumstances and a modification occurred within 2 years prior to filing.

As far as service, sometimes courts will not schedule a hearing until service has been accomplished. In such a case, obviously, you will not be required to go to court until the mother has served you. In some counties, however, such as Fulton County a hearing is scheduled as soon as the case is filed. If you are aware of the hearing, and therefore the filing, it may be a good idea to appear for the hearing, assuming you are a resident of the county in which the case was filed. If you are not present, and the mother has evidence to show you are purposefully avoiding service, the court could proceed without you and/or rule against you. I would suggest you consult with an attorney as soon as you are served or are aware a new matter has been filed to ensure that you do not waive any rights.