MWM, you said, "Wanting to stay in your room and sleep are signs of clinical depression..."
I agree, but given her shifts it could be that, too. Mind you, having a sleep pattern out of kilter can also lead into depression or exacerbate it.
Kjs, he's a rebellious early teen. He doesn't want to feel he has to always be in your pocket. They do this, especially boys. They distance themselves emotionally. And you have to let them do this. If you try to cling they pull away more. If you let go, they're more likely to come back to you because they want to, not because you've still got your apron strings tied round their neck.
OK, he lied to you. They do this. It sounds to me like he was lying about a lot more than just being online. It also sounds to me like he considered it a small lie to say he'd been watching TV because he knew you wanted him to talk online and he didn't want to. Yet he chose to lie to you rather than hurt your feelings about it? Actually, a good sign, from your point of view. It means he's not always trying to do his darndest to hurt you.
I suspect he didn't want to go to school (all that overdue homework is a big reason) because surely not all those people would be ill! Besides, to find out that all those people would be unavailable to get him to school would have taken a lot more organisation that I suspect he was doing. If you're trying to get to school you don't play computer games and wait.
Kjs, you have a very stressful life. Your family, your job and also how you are handling it. If you learn a better way of coping emotionally you will still have a stressful life but you will not find it drags you down so much. And when you're not dragged down so much it makes it easier for some problems to be managed at least a little, which can reduce the stress a bit more.
You sound like you're going to pieces so fast people are getting hit by the schrapnel; while your husband is doing the opposite, almost reactively. he's almost going overboard in trying to balance your reactions, and staying as detached and apart as he can. neither is healthy. I don't think you have a lot of control over your actions right now (exhaustion can do that to people; add in depression and it becomes uncontrollable).
How are you coping at work? Are you holding it together there? Is work a respite for you? Or is it dragging you down there as well?
Esther was right, you need some "me" space in your life, probably apart from your job. Instead of crying in your room, go for a walk. If it's daytime, wear big sunglasses so nobody can see you crying. Pretend you're Paris Hilton incognito. Even if it's 4 am, take yourself for a walk. Take pen and paper with you (and a torch, if it's dark).
And take yourself to your doctor and talk about how you're feeling. I do think you need help for yourself, so you are better equipped to help your family.
You keep saying that your family doesn't need you, you may as well be gone and they probably wouldn't notice. If you really believe that is true, then it should be easy for you to take some time and walk away so you can help yourself for a little while. But I suspect you know it's not true, which is why you're driving yourself into the ground like this. You feel like you're watching a train wreck in slow motion.
Maybe you are. Maybe it's time for you to run and get help. It's not one person's job to stop a train wreck. And the rescuers have to be in peak condition, or they only end up hampering the whole rescue effort.
Get help for yourself. If it means they flounder for a while, so be it. Maybe they'll have a healthier respect for all you do, if for a short time you're less available to do it. Do not let anyone dump blame on you (especially not your family).