https://www.tga.gov.au/covid-19-diagnostic-tests-included-ar...
Advice re Legal supply of COVID-19 diagnostic tests in Australia
https://www.tga.gov.au/legal-supply-covid-19-diagnostic-test...
Quote from TGA: "NOTE: In Australia, the supply of self-tests for most serious infectious diseases, including self-tests for COVID-19, is prohibited under the Therapeutic Goods (Excluded Purposes) Specification 2010. Testing for serious infectious diseases is best conducted in conjunction with a healthcare professional who can provide appropriate advice and treatment if required."
I can see it right there in the table. So it's definitely approved. On the self-test bit, of course not, even the manufacturer stresses this is for professionals to use. Not lay people.