Translation:Dad cooks dinner while my brother is going home.
I understand that "идёт" may directly translate to "is going." But, in this context, doesn't it indicate motion toward the speaker, such that "is coming" should be accepted?
I think you're right and it would sound better with "is coming" but I also think this translation just sounds awkward in general. I believe it's more common to say something like "Dad cooks dinner before my brother gets/comes home"
But maintaining the "while" sentence structure, "Dad cooks dinner as my brother is on his way home."
I don't understand while there are two tenses in the translation. I'm not an English native speaker, but for me it would be more logical as follows: Dad is cooking dinner while my brother is going home. Dad cooks dinner while my brother goes home.