1. Yes, a dependent or even a student can lodge an EOI, apply for PR
2. Depends on the Students course, if it's a bachelor or below dependent can work only up to 40 hours per fortnight. If its Masters then dependent can work full time once the studies commenced.
It could be hard to find permanent fulltime opportunities for a dependent even they have full-time work rights just due to the nature of their visa, most of the employers are not aware of dependent work conditions so they would reject applications as soon as they see the 'Student' visa. But if you can convince the employer, then it could be your lucky day