Is it obligatory to repeat the prayers that were performed with the prostration of forgetfulness (sujūd al-sahw) with the takbīr (saying "Allahu Akbar") when going down to prostrate instead of sitting, and is the prostration of forgetfulness combined if it is due to two obligatory actions, or is there a prostration for each action?

The manner of the prostration of forgetfulness is like the manner of the prostration in prayer: it consists of two prostrations separated by a sitting. It is sunnah to perform iftirash (a specific sitting posture) and tashahhud (recitation) in them. The takbir (saying "Allahu Akbar") in the prostration of forgetfulness begins with the start of the transition and encompasses it. If it is delayed slightly but is uttered before reaching the posture being transitioned to, then the prostration is valid. If the worshipper misses two or more obligatory acts, then one prostration of forgetfulness is prescribed for all of them together.