Hi.
I went through the process of getting 457 visa twice. Each time immigration agent was involved. First time it took about 8 weeks, second time it was only 2 weeks.
From memory I had to do chest X-ray each time. I think there was more thorough medical examination the first time I applied for that visa.

As you are not entitled to Medicare you can claim medicare levy when you do your tax return. you just need to get a special certificate from Medicare, which is very easy to get.

Private insurance is probably about $150/month. Check medibank's webiste to get quote