Are there any legal impediments to reciting the verses (At-Tawbah: 17-22, An-Nur: 36-38, Al-Jinn: 18-23) alternately after Al-Fatiha in the prayer of 'Tahiyyat Al-Masjid' (greeting the mosque), given that they relate to the nature of the prayer?

Committing to recite specific verses in the Greeting of the Mosque (Tahiyat al-Masjid) and consistently doing so is not from the Sunnah. There is nothing reported from the Prophet (peace be upon him) that indicates this. If it were legislated, he would have done it. Many innovations begin this way, then become a followed Sunnah. Therefore, adherence (to the Sunnah) is obligatory, and innovation is not.