Someone who is depressed isn't going to be able to 'make themselves happy' just by leaving uni/getting a job etc. He will eventually, with proper treatment, get over the depression, but it's not something with an overnight fix.
He's being difficult to live with, but he's not being unhappy/lazy just to annoy you, even if he's playing on some of it a bit.
Running away from problems won't fix them, but he's probably got to such a point he can't work out how to start fixing them.
Stuff I'd consider:
It's a shared home ie. he's living with you, so there has to be some level of agreement about proper standards in communal areas of the home (like emptying overflowing bins, throwing away rubbish, washing dishes, washing often enough to not smell) etc. I wouldn't be surprised if he's been behaving in a similar way in his student accommodation and has pissed off his flat mates enough that he's happier with you.
Beyond that level of basic stuff, what else can you negotiate with him that's he's willing to do to contribute to the household? Houswork, cooking, DIY, waiting in for delivery people?
If he's willing to discuss the position with you - when did he last see his GP? Is in on an effective dose of medication? Is it the right medication? Does he need a referral to the CMHT?
Again re uni, if he's willing to discuss this - most (I think all, but wouldn't want to bet on it) universities have student counselling services. Would seeing a professional to talk about things help him? This is something he could consider.
His students' union ought to be in a position to help with difficulties with academic stuff for not attending complusory sessions.
If he's having difficulties with academic work, the disability service may be able to support him with mentoring/other services.
If he wants to drop out for the rest of the year and start again next September (keeping the option to go back is usually wise) the uni careers service may have some advice re job hunting in the meantime.
All of that stuff is likely to be completely overwhelming to someone who is depressed and will need to be broken into small steps/to be done with some support.