Currently there are no restrictions on work for international students in Australia. This means that you can work for unlimited hours before your course starts and also during the course. Note that this is temporary measure and will be reviewed by the Australia government.
You are allowed to live, work and study in Australia after graduating with a degree from an Australian institutions. You will need to apply for Post-Study Work stream visa that allows you to stay for between 2 to 4 years after you have finished your studies. The length of your stay will depend on the level of the qualification that you complete in Australia.
bachelor degree (including honours): 2 years
masters by research and masters by coursework: 3 years
doctoral degree: 4 years
You may be eligible to get another Post-Study Work stream visa if you have live, work and/or study in a regional area while holding your first post-study work visa.
You can also include your family members in your application for Post-Study Work visa application