Custody question

I left my ex when I was pregnant due to DV and him harming me while i was pregnant. The is still yet for an avo to be put in place permanently. I have my daughter now and she has my last name and he isn't on the birth certificate. My daughter is 4 months old and i havnt heard from him in relation to custody, maybe its cause im not calming child support from him or maybe his new girlfriend is keeping him occupied and his other daughter to his first women. Would i know if he has made some sort of move on custody? I have already made a move. Would he know that? My lawyer is hopeless at following things up with me so im looking for a new one at the moment. He can only contact me through his lawyer as the is an interim AVO. Which he did when i was in my third trimester asking to have contact with me as he wanted to know about the pregnancy and wanted to be present at the birth. My lawyer told him where to go. He tried to kill her as if i would let him near her.
I guess my question is would i know if he has made a move? And if he does can he ask to change her last name or do i have to agree? Im hoping we can just walk away and never hear from him again. I hate the thought of him getting time with my daughter after what he did and his new girl friend looking after her as she has just left school. I use to look after his fist child when she came on the weekends morning through to night and the thought of a just 18 year old doing that with my baby gets to me and having my daughter watch him treat her bad i know he would im his second so he wont stop. it is killing me!