Did the Prophet (peace be upon him) apply Tajweed rules when making supplication (du'a), such as lengthening the word "السماء" (al-Sama') in the supplication "In the Name of Allah, with Whose Name nothing whatsoever on Earth or in the heaven can inflict harm"?

It has not been established that the Prophet, peace and blessings be upon him, recited supplications with tajweed and tarteel. Tarteel is specific to the Holy Quran. It has not been narrated from the Salaf (pious predecessors) that they recited Hadith or supplications with tarteel, or chanted them like the Quran. Therefore, it is preferable to abandon that. The essence of supplication is to be performed with humility, submission, and earnest pleading, far from artificiality, pretension, melodic intonation, and forced tarteel.