Wife and kids dont matter. The calculators are generally right as you wouldn't be on a different tax rate or anything, would still pay the levies. You just wouldn't be entitled to any centrelink family or childcare payments and need to pay for schooling.
You only need private health insurance to cover you from grant to registering for medicare. As you are entitled to medicare you can get insurance for a month to fulfil visa requirements until you register. If uyou still choose to keep private health insurance yes you will pay because you will still be using medicare, private health simply tops up what medicare gives you. Cuts the waiting and gaps you have to pay for.
As for best it depends on what you want covered and for how long. Look on iselect.com.au to compare. They show all ranges including 457 cover to top of the range.