먼젓번 담임 선생님의 말은 죽어라고 안 듣던 말썽꾸러기들이 박 선생님 앞에서는 고개도 제대로 못 들고 수줍어했다.
= The troublemakers who never once listened to the previous homeroom teacher have turned bashful even to look teacher Park in the face.
What comes before 말썽꾸러기 is a clause functioning like an adjective, similar to a relative pronoun clause.
[[{먼젓번 담임 선생님}의 말]은 (죽어라고) 안 듣던] *말썽꾸러기들*
= *the troublemakers* [who would not listen to [what {the previous homeroom teacher} said] (even for a second)]
죽어라고 is a common intensifier adverb, like "to death" in the sense of "extremely" or "desperately".
It's a short for "죽어라!" 하고, like " as if you wanted to die / as if you wouldn't even stop at death", but it is mainly a loose intensifier. It doesn't mean death literally. 죽을 듯이 is a less common synonym.
- 죽어라고 공부만 했다 = I just hit the books like my life depended on it.
- 죽어라고 뛰었더니 일등을 했다 = I ran like there's no tomorrow and ended up winning the race.
- 죽어라고 일해 봐야 남는 게 없다 = Even if I work my butt off to death, I don't seem to get anything back.