Friend doesn't want to rent with me..

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Youve really answered your own question. save uo money, get your mental health under control, then either go and study or consider mobing out when you have a job. A few years more wont matter. Cant say what you should do anout your friend and whether you wnat to let it affect your friendship. I do see its awkward.
Things change. If you get more under control and her flatmate doesnt turn out then it might become an option again.

She's got a very good point - friendship in terms of enjoying a person's company, having a compatible sense of humour etc is very different from being suitable domestically. I know I certainly could never live with my best friend, our approaches to 'use by dates' and the washing up alone are completely incompatible and to try and force it would end the friendship - which is far more valuable.

Sharing a house with someone where it is a more distant, businesslike relationship is far more reliable.

Living with someone full time is very different than having sleepovers, it can be quite a strain on a friendship.. I had one of my friends living with me and it was the worst experience.. We fell out for a long time, until he moved out and then we were good friends again! Maybe she's genuinely worried it will affect your relationship?

To be honest it's possible she doesn't wanna become a carer for someone with mental health issues by living with you. It can be intense to live with someone with mental health issues, even mild ones... once you're living together you see and hear everything that goes on and you can feel like you need to be helping all the time. That's understandable. I'd imagine that the main reason is what she said, not wanting to ruin a friendship - again a good reason! Living with friends does put a strain on if someone doesn't pay a bill, uses too much electricity/heating, takes long showers, is wandering around while you're sleeping... etc there are so many ways to annoy a flatmate it's not worth putting a friendship through it.

Living with my dad doesn't help my mental health issues as he can be quite emotionally abusive. If I have to wait until I've resolved all my issues, I'll never move out.

Im perfectly capable of taking care of myself.

I'm not saying you'd ask for any support. However, if you look down, she hears you crying, she knows you've had a bad day but she wants to go out etc... she's your friend at the end of the day she isn't going to just ignore those things if she sees them.

Then you will have to move out and live on your own or with strangers. You have to figure out if you can afford it. If your friend was to move out with you she might have been wary of your mental health issues and not wnat to be respobsible for them.

Will you be going to Uni?
Maybe you could consider a hostel or some form of sheltered housing for young people?

I wouldn't expect her or anyone else to be responsible for my problems and she understands that.
I won't be going to uni. Might look into hostels. I'd be eligible for housing benefit so I'm pretty sure I could afford it.

I understand your point but I couldnt live with someone who doesn't know me as well as my friend does, and she's the only friend I have.

It's difficult to say without knowing you both, but is there any chance she may consider that you would want to spend all your time with her? As others have said, the joy of sharing with non-friends is that you don't have to be social all the time.

Im perfectly capable of taking care of myself.

I don't think so. She works and I have life of my own.

Then live by yourself.