Hi.. I know how you feel, so much of this was so similar to what i had before :
1. Yes, speak to your parents. Don't worry about coming it shows the struggle you are having. Say to them you feel the counselling will help support you in better managing and controlling the urges of self harm and I'm pretty sure you're parents don't want you self harming and so should be open to you getting the help to do this. There is stigma attached to counselling my family came, even when i have a general review with the doc it's why etc. But getting counselling is private and it's not abour getting a label of being unwell, they don't diagnose you it's just like a conversation with a trained person and you could say thatto them. It's in school I'm presuming or an out patient centre and so that can sometimes be better ..but yeah listen to them and answer according. If not then when you speak to teacher you could just say look i want to but my parents aren't too keen on it as they have a few worries about it and the teacher might be able to help address some of these as well.
2. Okay so when you move to sixth form they would get a report on like your academic, grades the statement and everything else which is like information thst would be passed on for every student. On this it won't say any sensitive information. Any other additional info that needs to be passed on regarding that individual is sensitive like SEN needs, safeguarding yes this needs to be passed on for your care but this is separate and would be sent to the relevant college person ie the safeguarding lead there and therefore this is separate to your academic stuff/admissions stiff. Sayibg from experience. However again, feel free to ask a teacher about this so they can reassure you further.
3. I understand how hurt it must have made your feel when she said she won't and sge did.the approach of promising confidentiality should not be used as frankly proffessionallls can't keep everything confidential and sometimes they do have to for duty if care. Schools In that also have a lower threshold as well than let's say doctors. She should have explained that she can't keep everything confidenti and there's may be times when she has to share some info with other people and perhaps given you an idea as to who buy should say that essentially only people that need to know will need to know. When this happened to me it really upset me and everything but i just went up after a couple of days once I'd processed it and had an honest conversation with them about how it made me feel and how i would've appreciated then being honest. Honestly is most important in these things!