If you take electives over the summer and are essentially 4 courses (12 units) ahead in the Fall, is it possible to get permission in the Fall to take courses that require registration in level 3 or above while you're level 2 or will they just make you fill the space with more electives?

It depends on the courses. They may or may not grant you permission - it depends if you meet the other prereqs, if there is room in the course (students in level 3 will get priority), etc. Also keep in mind that the 3rd year courses may conflict with required 2nd year courses.