Is it permissible to supplicate with Quranic supplications during prostration and after the Tashahhud, and is it permissible to supplicate and seek refuge while the Imam recites, after contemplating the verses?

There is no harm in supplicating during prostration or the tashahhud with supplications from the Qur'an. As for supplicating upon reciting a verse of mercy and seeking refuge upon reciting a verse of punishment, this is a matter of scholarly disagreement. Some scholars consider it a sunnah for every worshipper, in both obligatory and supererogatory prayers. Others restrict it to supererogatory prayers, while some consider it disliked.