Why is the answer: they "will" not come home today? instead of: they did not come home today.
Also, why is it not "hawakuji"- when the verb is monosyllabic, shouldn't it retain the "ku"?
The "correct" answer (as given by the sentence discussion topic) is "they are not coming home today". The sentence is negative present continuous, hence no tense marker is used. As they are not coming home today the English sentence "they will not come home today" is also a valid translation, despite being negative future.
In the negative, even monosyllabic verbs lose the ku-.
Your sentence, "they did not come home today" would be hawakuja nyumbani leo. The -ku- here is the negative of the past tense marker -li- rather than coming from the verb kula. Note that only in the present tense does the verb ending "a" change to "i" in the negative.