"the road going to the house" is the correct answer, but not "the way going..." or the more natural "the road leading to the house"? and when i skip answering i get "the road home" as correct answer?
Both "the way going to the house" and "the road leading to the house" are now accepted.
간 is the past determiner form.
- -는/은: present determiner
- -ㄴ/는: past determiner
- -ㄹ/를: future determiner
As for action verbs, only -는 is a present form, and -ㄴ and -은 are past forms.
Another example of the need for context. You could compose a sentence where 지브로 간 길 would indeed translate "the road home." He chose the wrong road home. 집으로 간 길 잘못 걸렸다. The road home was the last road he ever traveled. 집으로 간 길은 마지막이였다.
"He chose the wrong road home." would be "집으로 가는 길을 잘못 들었다." Relative tense is used in the Korean language.
A road cannot move; the road isn't going home (as in "the road is moving to the house itself"). The subject for "가-" is omitted. Here, "집으로 가는 길" literally translates to "the road along which one goes home".