When a type 1 diabetic reaches high school and is independent, do they still do everything via the nurse?

Varies with location. A variety of state & local rules guide the administration of any type of medication and or treatment. In some states self administered meds might be the rule while others may require parents to come to the school and do it. This is something you need to clarify with the local officials.
It Depends. There are probably diabetics younger than high school age who manage everything themselves. Part of the requirements of the nurse at the school, might be more the school's rule more than medical guidelines. If you are able to manage yourself, perhaps you can ask the school if you can't do it yourself with the nurse just providing 'stand by' observation and supervision.