well he sounds like a complete w@nker and you need to put your foot down. he is either in this little girl's life - or he is not...his choice - and it will be his loss if he doesn't bother.
sounds like you are doing great without him....take him to court and get the child support he is legally obliged to pay. you should get legal advice & demand that any child access is supervised to start with as he has never had a relationship with her...and she should not have to go with what is basically a stranger to her.
when she is old enough...she will decide - but for now - you are her main protector. good luck - he sounds like a deadbeat dad...you are doing well on your own - well done!!
I'm not sure what "disability lawyer" means. If its some kind of court-appointed lawyer cuz it's medicare, then i dont know.
If an attorney doesnt do what hes supposed to, get a new one. Depending on how bad the damage is, you can decide whether to pay him for any costs incurred (which are different from fees, which if he didnt show up are zero)