What is the ruling on being negligent concerning the Sunnah; and is what the worshipper did by prolonging the Fajr prayer with Surat "Qaf" considered following the Sunnah of the Prophet (PBUH), or does it contradict the Prophet's guidance as stated in the Hadith "Are you a tempter, O Mu'adh?" And what should he do with his family?

It does not appear from what you have mentioned that those individuals are negligent regarding the Sunnah; rather, they found it difficult to apply it, so they asked you not to burden people with it. Since you wish to act upon the Sunnah, you should encourage them to it, explain it to them, and urge them to apply it gently and leniently. If some of it becomes difficult for them, it may be permissible to omit some recommended acts in order to win hearts. So, be gentle with them until they become accustomed to good.