Well, so just to clear any doubt I went on the government page.
www.homeaffairs.gov.au/trav/visa-1/457-#tab-content-1 and it says
" Health insurance
You will need to have made adequate arrangements for health insurance to cover your period of stay in Australia and you must maintain these arrangements for the duration of your stay. You will be asked to acknowledge that you have done this in your visa application form. For advice regarding the level of insurance that would be considered adequate see: Adequate health insurance.
Your insurance must cover you and any members of your family who come or remain with you in Australia. "
and if you go on the link, it goes into more details:
If you cancel or terminated the health insurance you stipulated before applying for visa, the provider will notify Immigration
"By providing this letter to Department of Immigration and Border Protection you are also agreeing that may notify Department of Immigration and Border Protection if you cancel, or fail to renew, your health insurance policy, or if your health insurance policy is terminated. "
But as I said, there are different levels of health insurance, you can buy the cheapest .
Back to suburbs now.
Balmain is nice, Crowns Nest is also nice with a "village feeling" and a very nice bookshop (the Constant reader) and good Italian cafe (sparrow).
I like Mosman more than Balmain and love Balmoral beach, my DH loves Double bay but $$$$$ . Kirribilli is quite popular too.
This is how great Sydney is, there are plenty of nice places to live. It comes down to what you like to do .
And because it took me more than a year to find a great bookshop (one of the cons of Australia, the lack of beautiful big majestic bookshops) , let me point you to it straight away: Kinokuniya www.kinokuniya.com.au in the Galeries in front of Queen Victoria Building. No majestic in the European way but the best selection in Australia. I usually combine my trip to the bookshop to a lunch at the sushi train on the 1st floor (in front of Muji). As you can guess from my name (Stephen King + John Irving= KingIrving), books really matter to me.