I want to answer your questions respectively. Even if I am going to answer your questions theoretically, I am sure you can handle the process.
First of all, after your marrige acknowledged by the court, you can get the family registry.
Second, the question sperated to two parts. First according to your son, which there is no problem with him. He will get the turkish citizenship from you. However, for your wife, the process is a little bit different. You and her should live 3 years in this marriage. And you should prove you are living like a family. You and your wife prove it is real marriage not a fake one
Finally, as far as I know you need a court decision to change your birth date on your passport. After you get the decision from court, you could prove your birth date. In this last question I am not a hundred percent sure, you could ask this question to Turkish Embassy.
Kind regards.