given your age what will probably happen is that they will have a chat with your first with your parents in the room (you can ask to go in alone if you want to) and then they will ask your parents to go outside and have a chat with you alone - anything you tell them which isn't an immediate safeguarding risk (i.e. suicidal thoughts/plans, abuse) will be kept confidential (although if there is something you don't want shared it might be worth stating it explicitly so nothing is accidentally disclosed)
as for weighing, if you are being referred for an eating disorder they will want to keep a record of your weight, they may not do this on the first session but the likelihood is they will want a starting weight for their records... obviously they can't physically drag you onto a set of scales and they aren't going to refuse to work with you if you refuse to weigh yourself, they will understand that it is a very stressful thing for someone with an eating disorder and that you are probably not comfortable with your therapist yet - however they may have an expectation that you eventually agree to weigh ins, this is something you will have to find out from your treatment provider (where I was treated they agreed not to weigh me for a bit and but I was eventually told I had to comply or I'd be discharged)
as for what they will ask, they will ask you why you feel you should be seeing a therapist and what your view on your difficulties is, they may ask you about what goals you have for the therapy, they will probably ask about your history (mental health, physical health, abuse etc) and about your parents mental health history, they may ask you a bit about your life in general (e.g. education, friends, hobbies) and they will probably explain what sort of treatment they will be doing with you - they will understand your nervousness and I don't know if you have anything to disclose but they will not necessarily expect you to be making any serious disclosures immediately (e.g. abuse)