Around that time they started to use family names but at different times in different parts of Norway, so Pedersen for a woman instead of Pedersdatter is possible.
Weird though about the gender, but mistakes happen. But they needed a passport to leave, and a birth certificate ( baptism record) to get the passport.
About abandoning his family, he might have had a good reason, maybe there was absolutely no work, maybe he was tricked, whatever. Maybe wife had an affair. Hard to know.
My father abandoned us, and started working many places, and often at the end of the first month the employer did not pay wages so he had to leave and find another job somewhere else, he was tricked. (he worked as lab technician in small local photo shops, or general stores that had photo developing and printing services. That was in Sweden.)