What is the ruling on the funeral prayer if the Imam made a mistake and said three "takbirat" instead of four, then said the fourth "takbira" some time after the "taslim," and was the Imam's action towards the congregants correct?

The Imam's action is correct if the interval after the salām is not long. As for the congregants who did not follow the Imam, their prayer is invalid because they did not pronounce the obligatory takbīr. The fard kifāyah (communal obligation) has been fulfilled by the Imam and those who followed him in rectifying the deficiency, and the remaining congregants are not required to repeat the prayer.