How do I go about marrying an Australian woman with the visa waiver program and temporarily staying in the U.S.?

If you marry your fiancee and file an immigrant petition and adjustment of status application, she would not have to leave the U.S. and would not need any special dispensation to remain here while the application is pending. If she does leave the U.S. before receiving a green card, it will be very difficult for her to come back until she goes to the U.S. embassy in Australia for an immigrant visa interview. She will also forfeit the adjustment of status application fee and lose quite a few months while the system digests the need to switch her case from the adjustment of status process (USCIS, Department of Homeland Security) to the immigrant visa process (National Visa Center, Department of State). If she needs to go back to Australia before she can stay in the U.S. for the months the green card process will take, it might make sense to get married, and hold off filing the immigrant petition until you can bring her back on a K visa. If your fiancee leaves the U.S., she can also re-enter on a visa waiver. There is no specific time she would have to wait before re-entry. However, there is a delicate issue here: if USCIS will later feel that your fiancee re-entered the U.S. with the intent to marry you and take residence in this country, she will be charged with immigration fraud and denied a green card. If she plans to re-enter on a visa waiver, consult an immigration attorney before she leaves the U.S.

No unless you get married and file a petition within the next 19 days.

You can marry her while she is here on the visa waiver and she can stay. Once married, she can apply for a green card. The entire process takes only 3 months from start to finish if done right. She will be eligible to receive a work/travel permit as well (while she is waiting for the green card). There will be an interview at the local immigration office to make sure that you guys are the real deal. Please consult with an experienced immigration lawyer before finalizing your wedding plans. The timing of the wedding can be an issue as well.