My understanding is that you can work while you're on the visa, but re-read the Department of Immigration "Partner visa (subclasses 820 and 801)" page that sets out what you can and can't do. You might want to contact the Immigration Department (anonymously) to discuss this. One thing that is clear is that you should notify the Immigration Department if your relationship circumstances change.
Since you are able to work on that visa I assume you would also be entitled to hold an ABN. You can apply online for an ABN here: https://abr.gov.au/for-business,-super-funds---charities/applying-for-an-abn/apply-for-an-abn/ and get an instant response as to whether your application has been accepted. If you have a tax file number, I don't see why you wouldn't be able to get an ABN, there is nothing on the government website regarding citizenship or residency requirements for ABN holders.