All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Our answer to you will be summarized in the following points:
1- We have already issued Fatwa 84403 clarifying how to make up for missed prayers if they are so many, so please refer to it.
2- The stressed supererogatory prayers which are performed before or after obligatory prayers, the Sunnah of Fajr prayer and other supererogatory prayers, all of them are not obligatory prayers, so you are not obliged in principle to perform them at all, and thus you are not obliged to make up for them if you have missed any. Moreover, the scholars differed into two opinions about a person who had missed obligatory prayers whether or not he should perform supererogatory and voluntary prayers. Their views are mentioned in Fatwa 88217, so please refer to it.
3- We have already issued Fataawa 83600, 83877 and 89834 about the method of performing Tasaabeeh prayer, so please refer to them.
4- We are not aware of any prayer called the prayer of thankfulness. What is confirmed in the Sunnah of the Prophet however, is the prostration of thankfulness; for more benefit, please refer to Fatwa 83280.
5- We have not come across any supplication called the 'supplication of Tahajjud'. What is confirmed is the supplication of the Prophet which he used to say after Takbeeratul-Ihraam (saying Allaahu Akbar to start the prayer) and before reciting the Quran when performing the night prayer or any other prayers, this is called by the scholars as the supplication of starting the prayer; for more benefit, please refer to Fatwa 90050. This supplication is legislated for all prayers whether obligatory or supererogatory.
6- Finally, please refer to Fataawa 84248 and 82878 about following the Schools of jurisprudence and details of their foundation.