Is the prayer valid if the worshipper joins the Imam in the third rak'ah, then assumes it is the second and sits for the tashahhud, then stands up and catches the Imam in prostration, then the worshippers divide at the final tashahhud between those who sit and those who stand to complete the rak'ah?

What the latecomers did by performing the bowing (ruku') and joining the Imam in prostration (sujud) is correct, as they were delayed by an excuse. And whoever is delayed from the Imam by two or more pillars due to an excuse, such as sleep, forgetfulness, or crowding, should perform what he missed and follow him, and his rak'ah is valid if he is sure he will not miss the second rak'ah.

There is no problem with those who sat with the Imam until he said the taslim and then completed their prayer, and they do not owe any prostrations of forgetfulness (sujud al-sahw). As for those who stood up thinking that the Imam had stood for the fourth rak'ah until he said the taslim, they should have returned to a sitting position to complete their prayer from there. This is because if a latecomer stands up to complete his prayer, thinking that the Imam has already said the taslim, but that is not the case, then the most correct view is that he should return to a sitting position and then stand from it.

If they completed their prayer and did not return to a sitting position, then if they heard the Imam's taslim before reciting the Fatihah, their prayer is valid based on the opinion that permits continuing. The same applies if they repeated the recitation after his taslim. However, if the Imam had already said the taslim after they had recited the Fatihah, then that prayer must be made up.

Scholars differed regarding the validity of following the Imam if there is a barrier between the Imam and the worshipper that prevents seeing the Imam. The more preponderant opinion is that the following is valid if one can hear his voice or the voice of someone who hears from him.