Can I apply for the Dutch student residence permit myself, or does my university do it?

In the Netherlands the student residence permit is sponsor-driven. The university or university of applied sciences (HBO) where you are enrolled must be a recognised sponsor listed in the IND public register, and only that institution can lodge the application for you.

This means your first step is securing admission to a recognised institution. The school then handles the IND submission, which is why you usually deal with the university's international office rather than the IND directly when you first arrive.