"If she was illigitimate, would her birth have been recorded? Would her father have been named? Would her mother have been named?"
A baptism would have been recorded, usually only listing the mother. In early 19th century records the father would have been entered retrospectively, usually with a notation that the child was legitimated by subsequent marriage, but that is not common in the 18th century. 18th century legitimized children would be baptized with mother's last name or no last name and then later show up with a different, presumably the father's, last name.
Sometimes illegitimate births and baptisms were not done in the home community due to the associated stigma but in some other community where the family had relatives.