Just for clarification -
How does this child fit in the picture, legally speaking? S/he is not the legal child of the decedent at this point. The OP's daughter is going to need to find a way to have paternity legally established before any of those things apply. This is not as simple as being appointed administrator in order to protect the child's rights, as the child may have no rights until s/he is legally established as the child of the decedent.
I thought about commenting on that, but I figured the OP has enough on her plate right now.
My guess is that, if the deceased's side of the family does not raise some objection, then it won't be an issue with the court. If they do, things get uglier. But that's something for the OP/daughter to discuss with the attorney.
While it wasn't really germane, I wondered how the deceased and sister acquired the house. Kind of unusual that a brother and sister would acquire a house jointly that one of them is going to occupy. Maybe through inheritance? Maybe as a gift from the parents? That could greatly affect the deceased's family's attitude and actions. They could be considering the house as "family property." And the child as simply "the daughter's child" rather than a grandchild and niece/nephew. And the only logical action for them to take to retain the "family property" would be to contest paternity. That possibility is something that OP and her daughter have to be prepared for.