Is the sermon of one who recites the entire Sura Qaf, chanted and Tajweed-compliant, after fulfilling the pillars of the sermon, valid? And did the Messenger of Allah, may Allah bless him and grant him peace, do that, or did he interpret it or preach with it?

Your Imam's sermon is valid as long as he fulfills its pillars. It has been authenticated that the Prophet (peace and blessings be upon him) recited Surat Qaf in the sermon, but he did not limit himself to it. Rather, he would mention other things with it, such as his saying: "To proceed: The best of speech is the Book of Allah, and the best of guidance is the guidance of Muhammad, and the worst of affairs are their innovations."