Can only speak from my own experience.
Most schools would sort out administration issues before the first day of school; generally when the place is confirmed.
Father's name does not have to be completed on the birth certificate. At the time you are speaking of, if the mother was married, there was a legal assumption that her husband was the father of her child.
In the old form of birth certificate there was no place to record the child's surname.
If it is part of their enrolment policy schools can request baptismal certificates to ensure the children gaining preference are indeed of the required religion. In my experience, this only tends to happen if there is pressure on places being offered.
However, I don't believe either teacher or school would have batted an eyelid at a lone parent in 1990s Ireland.