Australian citizen unsure of validity of Green Card. Can I apply for an ESTA to travel?

The question of whether you have abandoned residence is subjective and considers all the factors of your situation, so it is impossible for anyone to tell you definitively whether you would be considered to have abandoned your residence or not.

At this point, you should assume you haven't abandoned residence and go travel to the US with your green card, bringing with you evidence of your ties to the US (including US tax returns, etc.) and evidence of what is forcing you to stay abroad for so long. At entry, the immigration officer will either let you in as a permanent resident again (potentially with a warning), if he thinks you haven't abandoned residency, or, if not, he will ask you to sign the form to voluntarily relinquish permanent residency. If he lets you in, you're good; though you should consider applying for a Re-entry Permit while you are in the US to have less trouble after your next trip abroad.

If he asks you to sign the form to voluntarily relinquish permanent residency, you have two options: 1) if you wish to fight it you could refuse to sign, and then he will temporarily let you in and give you a notice to appear in removal proceedings where you will try to convince the immigration judge that you didn't abandon residence, or 2) you could voluntarily sign the form to relinquish permanent residency at that point, and the officer will (probably) admit you as a visitor. Of course, relinquishing permanent residency doesn't guarantee that you will be admitted as a visitor; you can still be denied entry like all potential visitors, but given that you just relinquished your green card, it's a strong sign that you are not intending to stay permanently in the US very soon.

Why not apply for a reentry permit: https://www.uscis.gov/sites/default/files/USCIS/Resources/B5en.pdf

It seems designed specifically for cases such as yours - that you are a permanent resident who has had a family emergency which requires you to be abroad for more than a year but that you do not intend to abandon your permanent residency in the USA.

The only complexity is that as a commenter notes, you have to be in the USA to file for a reentry permit. If you can, I’d try to return ASAP and to carry as much documentation with you that shows you haven’t abandoned residency (current mortgage or lease, utility bills, tuition payments, paychecks, etc) and then if you plan on leaving again, filing for reentry just to be on the safe side.