He isn't your mate anymore. It's awful awful awful to come to that realisation, but he isn't your problem, his depression isn't your problem, that he has no friends and nowhere to go isn't your problem. Try saying that out loud in a room on your own, in your car, in the loo ..... "That's no longer my problem, your decision" and it'll come naturally to you when he bleats at you.
You do not leave, this is your home, your kids home. Get it clear in your head, he has made this decision, he wants to follow through on this decision, you need to put you at the top going forward. What is best for you.
I have vivid memories of not being able to sleep, barely able to function. My head was banging with actual pain / overload of emotions. Have you seen your GP? Mine was awesome, got me some anti-depressants and had a chat with me, listened to me. Later on I got some NHS councelling too (although the first councellor was rubbish - complained about one session where I left in floods and saw her Manager, who was completely awesome and made me feel brilliant after one session).
Mad as it seems now, colouring in books allowed me to feel less panicked, my hands doing something while my brain overcooked. Small meals and often - fresh air if you think you can manage a walk. You matter, your health matters, look after you as number one.