It's ok to drop है in negative sentences such as this. However most Hindi speakers don't and I advice you not to.
the supplied answer 'does not go to home' is bad english. 'does not go home' or 'does not go to the house or to the home'
"Go home" was listed as the correct answer for me, so presumably it's been fixed since you saw it.
Let's hope they don't mean he's homeless or an orphan. Let's hope they mean that he goes to his piano classes or tennis or something instead of home. Otherwise thatd just be sad.