Off the bat, people can change, but they have to want to change themselves. It doesn't sound as though the wants to, and as you've said the relationship has ended, you don't need him to either.
I'm a man, and this sort of behaviour from other "men" really annoys me!
Can't do the dishes well enough? - you know what a dirty dish looks like, and what a clean dish looks like... turn one into the other, so that you'd be happy eating something off it.
I'd honestly start just washing the things you and your son use, then leaving his stuff in a pile - if he's not going to do any washing up, why should you do his?
Can't stay awake during the day? - stop the gaming (you're 27?!) and staying up watching YouTube.
If I didn't have a job I'd be spending all my time with my son - has he had a job before? How did he manage his tiredness, if he did have one? If he can manage it for a job, why can't he manage it for his family?
Depression does affect people differently, but his sounds like an excuse. In fact I don't think he really has depression, I think he is depressed. But being up all night doing nothing of any worth, and sleeping the days away is really not good for anyone. Does he get any sunlight or exercise?
These are questions you should ask about a teenage boy, not a 27 year old "man". I think it's all moot anyway, as it sounds like you've made your mind up about it all!
My advice (apart from only doing your dishes from now on, at least until you move out!) just enjoy your son l, and stop even bothering with your "partner".
He's not bothering with you, or your son, and has decided that his nocturnal activities are more important than his family activities.
Good luck.
(p.s. How does he bathe him 7 minutes?! Do you undress him, run the bath, then literally just hand him over for the actual wash? It takes me that long just to run the bath!)