What is the ruling on the prayer of one who erred in pronouncing the opening takbeer (takbeerat al-ihram) and in pronouncing the tasleem, then intended the prostration of forgetfulness (sujood as-sahw) to compensate for the deficiency in the tasleem, while it was originally for something else? And does his prayer become invalid if he intends to repeat it, thinking it is invalid, then it becomes clear to him that it was valid?

The basic rule is that the takbīrat al-iḥrām (opening takbīr) and the salām (concluding salutation) must be performed correctly; any deviation invalidates the initiation of the prayer or the exit from it. The preponderant opinion is that only one salām is obligatory. What appears to you is merely a delusion from Satan, and you did well to ignore it. Your prayer is valid and does not need to be repeated. You must continue to resist these whisperings (wasāwis), and do not judge your prayer to be invalid except with absolute certainty.