Is it permissible to begin the Tarawih prayer before fasting, and is it a condition to complete the recitation of the Qur'an in it sequentially according to the Mushaf?

What the second Imam mentioned is incorrect. Ramadan begins at sunset on the last day of Sha'ban. If the crescent moon is sighted, then Ramadan has begun, and the Tarawih prayer becomes legislated for that night. It is not obligatory to recite the entire Qur'an in order in Tarawih, nor is it forbidden for the second Imam to start from a place other than where the first Imam stopped. However, it is recommended for him to do so, and it is disliked by some jurists for him to recite from a place other than where the first Imam finished. It is incumbent upon anyone who issues a fatwa declaring this forbidden to fear Allah and to restrain their tongue from issuing such fatwas.