To what extent is the intonation (tartīl) of Du'a Al-Qunūt legitimate, and is tajwīd not exclusive to the Qur'an but rather one of the characteristics of the Arabic language?

Recitation (tartīl) is specific to the Noble Qur'an. There is no command to recite other things, such as hadith or supplications, with tartīl. It has not been reported from the Salaf (pious predecessors) that they recited or chanted them like the Qur'an. Therefore, it is preferable to abandon that, especially if it causes confusion with the Qur'an for listeners.