"Dad cooks dinner while my brother is going home."
Translation:Папа готовит ужин, пока мой брат идёт домой.
In a previous statement, someone rabbitted on about how slavic languages need not use pronouns when it was "obvious" about ownership and in this very next question, Duo in their amazing style go against everything that was in the last comment. But hey, its free I must keep telling myself that
The English is very confusing. If the brother luves wgere dad is cooking, i woukd say while my brother is coming home. If the brother lives somewhere else, going home is fine, but why would dad be cooking in that period of time???? Will he stop cooking when the brother gets to his house???
Ходить is used for repeated trips, non-direct/erratic motion, motion in general, and motion to and back from somewhere. Идти is used for motion directly towards somewhere during a one-time trip. If you used ходить here, the brother would be taking multiple trips home and back or walking in circles or something along those lines. In theory, those options are maybe possible based on the English sentence, but they are unlikely to be the meaning of that sentence. I wouldn't use "is going home" for someone who isn't making a concrete, one-way trip home.
Many people seem to be asking you this same question. I think that "homeward" is probably a good translation of "домой" for them.
"I go home", home here is a noun, but "I go homeward", homeward here is describing my 'going', that is, it is an adjective, and most English speakers will see that it is not a noun.