What is the ruling on someone who pronounced the opening "takbeerat al-ihram" in the second rak'ah by mistake, thinking it was the third, and is prostration of forgetfulness (sujood as-sahw) obligatory upon him? And what should he do if he forgot the prostration of forgetfulness and did not remember it until just before Fajr?

It is not clear what is meant by your takbirat al-ihram (opening takbir) in the third rak'ah. Takbirat al-ihram is a pillar at the beginning of the prayer. If the extra takbir or dhikr was due to forgetfulness, then according to the majority of scholars, the prostration of forgetfulness is not required for it, and the prayer is not invalidated by omitting it. If you meant raising your hands when standing up for the second rak'ah, thinking it was the third, then there is nothing upon you, and the prostration of forgetfulness is not prescribed for you, and your prayer is valid.