1) Centrelink it will be you and your wife who will register.
2) If you are only here for 8 days I would probably wait as you all need to go in with your visa they then give you a temp number and they usually post your cards out 10-14 days later. Although having them ASAP will help you with 100 point checks when applying for rentals etc.
3) Do you need private health insurance as part of your visa, as you are on PR
i wouldn't think it is needed. You are moving to Adelaide and I have heard there are excellent specialists there so personally I don't think you need it. I never had it in Melbourne, in Cairns I do because we don't have enough specialists and I often have to travel for treatment. You usually find health insurance runs from January to December. If you do sign up you have a 12 month waiting period for any pre existing conditions.
4) Again you can do it online once you are here but this can take longer to come through than medicare so unless you have relatives whose address you can put down I would wait until you come back.
One thing I would do, if you open an account online with NAB they will have your bank cards ready for you to pickup when you arrive. The other banks you usually have to go in for them to see the ID then they order your cards so again you may not get them in time if you use a different bank.