Going by what I found online, it looks like
comenzar really are pretty interchangeable, with
empezar being more common in Spain and
comenzar more common in Latin America. Both mean "to start", but
comenzar is generally translated as "to begin" because (in Spain at least) it is seen as a bit more formal/literary.
Iniciar is more formal/technical than either, with "to initiate" being a decent translation.
"Tu madre" is a shorter form of a command inviting you to do something improper with your mother! It doesn't have to be alone and it doesn't need a special intonation to get you into trouble in the wrong places. In Mexico and elsewhere in Latin America, polite society can go to any length to avoid using even the single word "madre" in day-to-day conversation, preferring "mama" in its place. It is troubling that Duolingo is exposing us to this phrase, even though it steadfastly shields us from "bad words" elsewhere.