[Wanderer]

The 457 is a temporary residence visa and is reliant on maintaining a sponsorship whereas the 121/856 are employer sponsored permanent residence visas and the 121 is the offshore version so it may be the onshore version you should consider when appropriate.
There are two ways to qualify for seeking an employer to sponsor you and that is by waiting until you have been on a 457 for two years or if you have qualifications assessed and a minimum of three years experience you can seek to do it earlier, in either case you'll need the agreement of the employer who also needs to be eligible.
Details are @ Skilled Workers Permanent Visa Options- Employer Sponsored Workers - Workers - Visas & Immigration

While on a 457 you'll need to have medical insurance and you'll also be liable for school international student rate fees.

 

[Wanderer]

You will see under the 457 employee eligibility that health insurance is mandatory for an application to proceeed.
Skilled Workers Temporary Visa Options - Employer Sponsored Workers - Workers - Visas & Immigration
IMAN Australian Health Plans - Health Plans for Temporary Residents Working in Australia is a company that specialises in providing cover for 457 and other TR visa holders.
On a 457, once you are in Australia and enrolled in Medicare that is sufficient to meet eligibility conditions, thus

Being enrolled with Medicare under reciprocal health care arrangements is sufficient to meet the health insurance requirement at visa grant and to comply with visa condition 8501.

You can only enrol with Medicare if you are in Australia. If you have enrolled with Medicare you should provide evidence that you have been issued with a Medicare card as part of your application.

If you are overseas, you must arrange adequate insurance for your initial period in Australia and provide evidence of this insurance as part of your application. You may be eligible to then enrol with Medicare once you are in Australia. Being enrolled with Medicare is sufficient to comply with visa condition 8501.

You should also be aware that once you have enrolled with Medicare, you need to contact the Australian Taxation Office to see if you will be subject to the Medicare Levy Surcharge. This might affect the type of health insurance policy you need to maintain while in Australia.

from the 457 H.I. FAQ
You can see the reciprocal details @
Visitors from the Republic of Ireland and New Zealand - Medicare Australia

You should also note that the 856 is optional as far as being offered by an employer and there are slightly different eligibility criteria for them as well as for an employee.