Everything is inspected to assess any potential harm to the environment. There should be no problems with importing wooden furniture and mattresses. However, only customs staff can decide that. There should be no taxes on personal effects, including electrical appliances.

For further, more detailed and accurate information, please refer to the relevant government website (below.)

https://www.australia.gov.au/information-and-services/passports-and-travel/customs-and-quarantine