There's lots of practical advice on this website
Tenants - NSW Office of Fair Trading

ETA - TBH I think that you might have to choose between being all buddy-buddy and standing up for your rights. It sounds like your ex is just palming a bunch of junk off on you. Old mattresses are for pets and he can have the TV from the pantry.
It might be worth contacting legal aid or the child support office to start sorting out custody and financial issues. If you have an advocate to discuss these things for you it can result in a better outcome because they don't have the emotional attachment or the fear of being the ***** that women often have.