1. You’re eligible to apply for a specific birth certificate where:
(a) It’s your birth certificate;
(b) You’re a parent (i.e. shown on the birth certificate as such);
(c) You’re their legal guardian/carer;
(d) You’re a member of their family (e.g. spouse/partner, sister/brother, son/daughter, grandparent/grandchild, aunt/uncle, etc.);
(e) Acting on behalf of a financial institution, executor of their estate, legal representative etc. (and provision of reason why you’re applying).
Proof of ID must be given with the application, including substantiation of relationship (e.g. document establishing guardianship, their written authority, power of attorney).
The Registrar may consider other applications, but they would be an exception. Unless you’re one of the above, I’d suggest you’d need an order from the court as part of a proceeding (and even then, the certificate would likely form part of the evidence the court would expect anyway).
2. An application to register a birth must be signed by both parents. As stated on the form "If the parents are separated or are in dispute over the name of the child, both are still required to sign the application as both have rights and responsibilities to the child, even if they are no longer in a relationship. The registry will make further enquiries if it is not satisfied with the information or the explanation as to why only one parent has applied." I know that’s not an answer, but it’s a start.
The Registry will register only the mother on the birth certificate where the father is unknown/unco-operative. Father’s can be added later (see next item).
3. The Registry won’t add a father to a birth certificate unless both the mother and father sign the form to do so, or there’s a court order. So, in short, not without the father’s knowledge on some level unless the court is satisfied it can make the order without his appearance.
4. Removal of a name is considered a ‘correction’ of the notice. While the Registry may correct minor mistakes, I would suggest that a removal of a parent can only be done by court order – and that the court would be looking for some appearance from the father unless there are sufficient grounds to proceed otherwise. There would have to be substantial evidence to prove he is the father, and compelling reasons why he won't participate in the hearing.