Is it permissible to repeat the prayer because of the incorrectness of the Imam's transitional takbeers and the haste of the congregants in performing the pillars?

The place for the takbirat al-intiqal (transition takbirs) is during the transition to the next pillar (rukn). Violating this does not invalidate the prayer according to the majority of scholars, as the takbir is considered a sunnah (recommended act) by them. However, according to the Hanbalis, the established view in their madhhab is that the takbirat al-intiqal are obligatory, and whoever intentionally performs them after the transition, their prayer is invalid. Nevertheless, some verifying scholars among them have favored the view of non-invalidation. As for someone who omits them or performs them out of their proper place due to forgetfulness or ignorance, they perform the sujood as-sahw (prostration of forgetfulness) and their prayer is valid.

One should not abandon praying behind an imam who performs the takbir or tasmee' (saying "Sami' Allahu liman hamidah") out of its proper place. Rather, he should be gently informed and taught the Sunnah.

However, praying behind an imam who does not achieve the obligatory level of ṭuma'nīnah (tranquility/stillness) in his prayer is not valid, because ṭuma'nīnah is a pillar. The ma'mum (follower) must then separate from him and repeat the prayer if he prayed with him.