Is the prayer of one who has completed a supererogatory prayer valid, and is the opening "takbir" counted for him, if he entered the mosque and found the congregation had finished, then prayed with another congregation a short prayer, then repeated the prayer in full with a third congregation?

We congratulate the questioner for his eagerness to attain the opening takbir (takbirat al-ihram), due to its merit, as it is authentically reported that whoever prays to Allah for forty days in congregation, catching the first takbir, will be granted two immunities: immunity from Hellfire and immunity from hypocrisy. If the questioner's prayer became invalid after performing two rak'ahs behind a traveling imam, then invalidating an act of worship is forbidden, for Allah Almighty says: "And do not invalidate your deeds." (Quran 47:33) Changing the intention of a fard (obligatory) prayer to a general nafl (supererogatory) prayer is permissible. Making up the invalidated prayer behind another imam is sufficient, and we hope that the questioner has attained the reward of the opening takbir with the new imam, because the mentioned reward is achieved by attaining it with any congregation.