What is the authenticity of the hadith "Do not precede the Imam and do not fall behind him," and do its narrations include the phrase "do not fall behind him"? And is the prayer valid for someone who prolonged the last prostration, then caught up with the Imam in the Tashahhud, and greeted (said Salam) with him or a few seconds after him?

Competing with the Imam in prayer is forbidden, due to the Prophet's (peace be upon him) saying: "Does he who raises his head before the Imam not fear that Allah may transform his head into the head of a donkey, or change his image into the image of a donkey?"

As for lagging behind the Imam, there is no explicit text prohibiting it, but the Sunnah is immediate follow-up to the Imam, based on his (peace be upon him) saying: "When he says 'Allahu Akbar,' then say 'Allahu Akbar.' When he bows, then bow. When he says, 'Sami‘ Allahu liman hamidah (Allah hears those who praise Him),' then say, 'Rabbana wa lakal-hamd (Our Lord, and to You be all praise).' And when he prostrates, then prostrate." The "then" (fa) here implies immediate sequence.

Delaying in a pillar (like someone who delays in prostration to make supplication) does not invalidate the prayer, but it is contrary to the Sunnah, and it is better to follow the Imam immediately.