What is the correct number of Tasbeeh, Tahmeed, and Takbeer after the obligatory prayers? And what is the wording of the remembrance before sleep and its reward? And how is Tasbeeh done with the right hand?

The prescribed remembrance (dhikr) after prayers has various forms: among them is to glorify Allah (subhanallah) 33 times, praise Him (alhamdulillah) 33 times, exalt Him (Allahu akbar) 33 times, and complete the hundred by saying "La ilaha illallah wahdahu la sharika lah..." (There is no god but Allah, alone without partner...). Another form is to say subhanallah, alhamdulillah, and Allahu akbar ten times each.

As for the remembrances before sleep, they include glorifying Allah 33 times, praising Him 33 times, and exalting Him 34 times.

The Sunnah for counting the remembrances is to use the fingers of the right hand, preferably the fingertips. There is nothing specific regarding which finger to start with.