Is it permissible to combine error and correction in the Takbir (Saying " Sami‘a Allahu liman hamidah " then " Allahu Akbar ") while moving to prostrate in prayer, then perform the prostration of forgetfulness (sujud as-sahw) after the taslim? And is it obligatory to stand and say the Takbir before descending for the prostration of forgetfulness?

You mentioned a prescribed act in the wrong place, which is saying "Sami' Allahu liman hamidah" (Allah hears those who praise Him) instead of "Allahu Akbar" (Allah is the Greatest) for the transitional takbir. You acted correctly by performing the prostration of forgetfulness (sujud al-sahw), and the description of the prostration of forgetfulness that you provided is the prescribed one. As for standing up to pronounce the takbir in this situation, we have not found any scholar who advocated it.