Is anything required of someone who forgets during prayer, yet reminds himself that he is in the Dhuhr prayer or in such-and-such a رک’ah?

What you are doing by ignoring the waswasah (whispers of Satan) is Islamically required and an effective cure. However, reminding yourself of the prayer and the rak'ah you are in is not advisable and will increase the waswasah. The person afflicted with waswasah is relieved of hardship in many matters until the waswasah disappears. So, if they doubt the number of rak'at, they should base their count on the higher number. And if she doubts the prayer itself (whether it is Dhuhr or Asr) and remembers before doing anything, her prayer is valid. But if she acts while in doubt, it is invalidated according to many scholars.