I think you should put him down with csa because you have told births deaths and marriages you know who he is and with centrelink if you know who the father is but don't try to claim child support they majorly reduce your payments.

It won't force anything to happen as far as custody goes. your ex would have to go through the court to get custody so there is no real down side I can see for you.

Also as far as your house goes you can actually go through court and they can force him to sell. I would definitely definitely see a lawyer about this contact legal aid. My dad was abusive and when mum moved out he lived in the house for several years and wouldn't sell it, he eventually agree to sell it after someone put a letter in the mailbox to offer to buy it, mum had seen a lawyer to find out what she could do (he didn't know about that) he was paying the mortgage at the time and even though he wasliving there and wouldn't sell legally he could have tried to get money back off her for the rent once the house sold and there were other downfalls, so I would really get onto that also you would be entitled to a higher percentage of the money from the house due to you caring for your dd.