I'm not sure how the system works in your area, but generally you will be asked for a next of kin, but as you are an adult it is possible they might not inform your family against your wishes. If you don't answer questions it is more likely that your family will be contacted whether you want them to be or not.

If you refuse to answer questions then it will be hard for the crisis staff to assess your risk and if they are really concerned then they might decide to detain you for your own safety. The best thing to do, is to be open and co-operative, put the emphasis on the fact that you are asking for help because you want to be safe and that you want to work with them. Too many questions can be overwhelming, don't be afraid to let staff know that you are finding it hard and want to take a break from all the questions.