Does the Imam's sutra remain a sutra for those who are catching up on missed rak'ahs after the Imam has said the tasleem, or are they required to take a new sutra? And what is the ruling on people passing in front of those who are catching up on missed rak'ahs?

If the Imam says the taslim and a latecomer stands up to complete what he missed, he is then praying alone. He must prevent anyone from passing in front of him, because the Prophet, peace and blessings be upon him, commanded this. Some people may neglect to prevent passersby out of ignorance of the ruling, or thinking that they are still considered to be praying behind the Imam.