Can you carry duty-free items purchased in an Australian airport into the U.S.?

Answer 1

Yes, certainly. Then you declare them as the imports they are and pay whatever U.S. duty the nice customs officials require, if any, as well as whatever "sales and use tax" your state requires for things you import (usually the same tax as if you'd bought them at home, minus whatever sales tax you paid when you bought the item).

"Duty free" does NOT mean you don't pay duty, only that the Australian retailer didn't pay import duty, thus passing along the savings to travelers who may be exempt from such duties for one reason or another (not including smuggling), such as a "duty-free limit".

This confuses a great many travelers, as does the "tax-free shopping", which simply means you don't pay sales tax to the retailer, but you still owe tax directly to the tax collector when you take the items back (or receive them by mail or common carrier) to a taxable state where you reside. Tax-free only applies generally to residents of such states, not visitors from sales-tax states.