If you do get diagnosed with a mental illness, it doesn't change who you are. A label changes nothing - you are the same person you always have been. It's illegal for a workplace to discriminate against someone due to a mental illness, and it is very, very unlikely that it would affect your ability to work or your university. Application forms may ask about mental health problems, but they cannot discriminate against you for that - the questions just mean they can put in place some strategies to help you if you need that.
Seeing a doctor/counsellor/someone does sound like it might be worth a try - maybe they could help you find some ways to stop self-harming? Is there any way you could talk to your parents about what's going on? Or even just tell them you're going to the doctor for another problem, and go without them?
Good luck
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