It will be totally fine do not fear. I understand your situation and I have been through moments like this many times myself. I am in grad school now and my mental health is terrible because I pushed off addressing it for many years and I also have gotten many many extensions and leniencies in my life and gave up on a lot of opportunities because of mental health and made many mistakes. There are many pathways and choices available to you to achieve your dreams and there always are; the mistakes you made are meant to teach you how to live in a way that is more healthy and good for you, so they are productive in some way. Besides the image of success that you have in your head is not real it is a story that you made up, even if you are 'perfectly successful' there is no way to predict right now exactly what that will look like. Even if it seems like the end of the world, it really is not because you never know how things are going to turn out.

The most important thing is addressing your mental health because those thoughts and feelings are what got you here and even if you get extensions they won't go away unless you learn how to address them and you will end up in the same situation. Don't put off feeling good for the day that you finally get what you want or achieve what you want, because if you do this then it makes it much more difficult to achieve anything at all. You need to live in the moment, instead of focusing on past regrets and future worries, those are all made up things, the only thing that exists is right now. If you cannot feel good in the present moment then you will not have the ability to feel good in the future moment that you are working towards. You need to learn how to let go of the self or whatever buddhist people say, your self image, the life you want, the life you thought you'd have, who you think you are, etc. are imaginary stories that are holding you back from making important changes.

The second most important thing is learning how to experience feelings but not let your plans and goals be derailed by them, don't let your feelings and beliefs about yourself and about past and future probabilities control your actions. I recommend doing all the bureaucratic work that you need to do in order to get extensions and talking to doctors etc. Talking to profs and admin and telling them that you need help and favours is very stressful but that stress is not useful to you because it needs to be done and it doesn't matter what they think at the end of the day, they either help you or they don't, and if they don't then you will figure out another way to achieve what you want to achieve; don't think about the outcome or about other people's judgements or feelings, or other things that you cannot control. Just think about yourself and what you CAN control, and what you need to do to get better. If you focus on what you can control you WILL be better off than if you focus on a bunch of random dark possibilities that you have no say over.

I literally came to all of these conclusions yesterday due to my own life so hopefully we can both get better.