I have 2 questions related to prayers.

1) For optional prayers that are prayed in units of 2, 3 or 5 rak'ahs, should the opening supplication be made between the takbeer and recitation of Surah Al-Fatihah?

2) In my area, it is common that people recite Surah Al-Fatihah as part of their Dhikr after the prayer.

Is there any basis for this (authentic or inauthentic)?

If the salaah of three rak'ahs is made in a continous form without a break or salaam after two rak'ahs, then there is no need to recite the opening supplication in the third rak'ah because the salaah is continuous. However, if there is a salaam after two rak'ahs and a person gets up to pray the remaining rak'ah as a single unit like in the case of witr, then he or she can make the opening supplication.

As for the case of reciting Surah Fatiha after salaah that is a clear bid'ah because there is no evidence for that. That was not the practice of the Prophet (ﷺ) or the Sahabah. Therefore this action has no basis.