What is the ruling on intoning supplications (du'a) within and outside of prayer? And is the prayer of one who performed Tarawih behind an Imam who intoned the du'a in Qunut valid?

The fundamental principle of supplication (du'a') is that it should be based on humility and submissiveness. Paying attention to blameworthy melodic recitation (tan'gheem) in it is disliked, because it distorts its intended purpose, which is supplication, turning it into play, as mentioned by Al-Kamal Ibn Al-Humam. As for reciting with tarteel (measured, melodious recitation) and adhering to the rules of tajweed (definition) (elocution) in du'a' without tan'gheem, scholars have differed on this. Some have prohibited it due to the lack of transmitted reports, while others have permitted and even encouraged it, considering that tajweed is among the beauties of the Arabic language and is not exclusive to the Qur'an. This is what Al-Qasimi alluded to. The matter is broad, and the prayer is valid in all cases.