My children have dual citizenship US-Australia. Can we travel to Australia on US passports and ETAs?
A passport is the best way to show that you are an Australian citizen. You might still be able to enter Australia if you are an Australian citizen without an Australian passport, but it will be more difficult. The airline might also stop you from boarding a plane to Australia.
Also, from faqs on the ETA site:
I am an Australian citizen with citizenship with other countries as well. Do I need an ETA to visit Australia?
Australian citizens have an automatic right of entry and do not need a visa or an ETA to enter Australia.
So, none of that says you cannot use an ETA on a foreign passport to enter Australia as a dual citizen, but you may face difficulties in doing so.
Probably not. As Australian citizens, your children are not eligible for ETAs, and they not will be allowed on the plane without one. If they do make it to the Australian border, there may be some hassle while they figure things out, and more hassle if leaving the country without an Australian passport (although you could apply for one while in country to avoid this).
That said, some people have reported applying for ETAs as Americans and (incorrectly) being granted one anyway, so your mileage may vary.