I haven't read the webtoon and I think this might just a plothole but here's my guess:
since Haru isn't a character at first and then just an extra, the writer couldn't be bothered to write a background for him (no parents, house etc.). Haru, unlike other figurants does have an ego, so he doesn't just mindlessly leave the school grounds at night, but stays there. So, yes he sleeps at the school.
In episode 10 (party episode) LDH mentions that he made Haru wear a suit from his brother, so everything he wear that isn't his school uniform is borrowed, until he becomes an extra and the writer draws his outfits for him.