Hey. First of all, I'm so sorry that I wasn't able to reply to this yesterday when it came in. If you did speak to your mum this morning (if it is this morning for you), then I really hope it went okay.
You are not a failure. Uh-uh. Nobody is a total failure. You can not be successful at things, but hey, nobody's perfect!
It's not too late for you to go back to seeing the psychologist. Perhaps your mother will be more understanding now that she's seen how things are affecting you? Whether your mother approves of it or not, if she's not going to go as far as downright forbidding you to see him, then I think that you should do what you want. It's your life you're living and your life you have to make better. Sooner rather than later is a good thing to bear in mind: why put off until tomorrow what you can do today? A psychologist might also help you to resolve some of your issues surrounding college and whether or not to retake.
If you had continued getting help, then perhaps things would have been different, but perhaps they wouldn't. The only thing you can do now is to try and put them as right as you can from where you are at the moment. Things could be a lot, LOT worse: at least you still have a chance to get that scholarship. It's not out of reach: it's just on a shelf a couple of inches above your head. Where you are at the moment is important: how you got here isn't. Learn from your mistakes and then forget them. They're not important any more.
When something bad happens, all anyone wants to do is for it to go away. That's why you want to sleep: it'll go away. College will go away, grades will go away, cancer will go away; everything will go away. If you let everything go away, though, then sooner or later, it's gone and you'll want nothing more than to have it back. And if everything goes away, then you're only left with yourself, and that means thinking things you don't necessarily want to think.
I know that nobody in your life has died, but I'm going to show you this anyway: the stages of grief. I think perhaps that they apply to you, with the thing that's died being where you wanted to be today. You've lost the scholarship and that's knocked all the cards down: cancer, the psychologist, everything that's happened since January. Losing the scholarship didn't cause those feelings: it's just brought them to the surface again.
* Denial (this isn't happening to me!)
* Anger (why is this happening to me?)
* Bargaining (I promise I'll be a better person if...)
* Depression (I don't care anymore)
* Acceptance (I'm ready for whatever comes) 
It's hard for me to see exactly where you fit in there, but I think that I can see parts of all those five things in your post. Maybe that's something for you to think about.
Take things slow. Okay, so you have to repeat two classes? That's not so bad: it could have been a lot worse. And if you do repeat, then you won't have to do nothing but work: you'll have time for other things too. You might find new people in the classes you retake, which could help you with your social skills anyway.
Personally, I think you should retake: you've got such a great opportunity within your reach. If you don't think it's the thing for you to do, though, then nobody - not me, not your mum, not your teachers, not anyone - should make you do it. Just make sure you know where you're going after this summer.
As far as your father's cancer is going, I think you should talk to him about tit. It's just eating you up inside. If you speak to him, then at least he will know that it is affecting you. If you bottle things up inside, then eventually you explode. If you open your mouth, then like popping the cork, some of the pressure begins to release. You shouldn't try and take all of something when you don't need to.
It's clear that you've had a rough semester, but you need to put that behind you now. Look towards the future. Talk to your dad, speak to a psychiatrist, work out where you're going with college and then work that out with your mum. Keep looking forwards: you don't need to look back. You are where you are and that's all that matters.
If you want to talk, then I'm always here, mmkay? Just message me and I'll get back to you as soon as I can.
I really, really hope that life begins to look up for you.
-Fern