I am also Australian and would like to help you find some clue as to where you can go for information on your ancestors. In fact, the Mecklenburg side, if your great-great-grandfather was born in Mecklenburg-Schwerin, should be quite researchable as the Mormons have filmed the church registers of that area extensively. But it takes time to get the films from the USA and October might be cutting it too fine.
On the National Archives of Australia site I found a naturalisation record for an Anton Schultz from 1853 - unfortunately it gives no clues as to place of birth. But do have a look (if you haven't already). The URL is:
Do you have a town of birth at all??? Have you accessed his death record or marriage record at the Public Records Office in Melbourne? These might give his place of birth. Without this you would not be able to get any further, because Schultz was and is a common name in Germany.
Have you seen this site?:
If you scroll down to "Family History" you will see that you can access this kind of family information online and even get uncertified copies digitally. It costs very little money to do this and would probably be worth your while.
If you can find a place of birth of Anton Schultz I would be glad to advise you from there.
In terms of your great-great-grandmother, "Hamburg" as a place of birth is also too vague. Hamburg was already a big city at that time. If you can search the online births ,deaths and marriages for her also, that may help you place her in a particular part of Hamburg. Otherwise it is possible to write to the big archives in Hamburg and ask for a search of old church records there - that will be expensive, however. Let me know if you would like me to search out the addresses for you.
Have you checked to see if any other family members have done this research before you? Many people put their research on line now, and there may be a distant cousin out there somewhere who has already done it all. Sites such as Ancestry.com and Rootsweb and One Great Family (although American-based) have many family trees that are accessible on line (for a fee). Or just Google your people and see what happens.
Hope that gives you some starting places. Let me know what you find, if anything. I am happy to help if I can.