Translation:He doesn't eat breakfast.
This sentence, depending on the context, can have at least two interpretations: (1)He doesn't have a habit of eating breakfast. (2)He isn't going to eat breakfast, maybe because he gets up late or he feels sick.
Historically, 他 was gender-neutral. It's only since the early 20th century that 她 started to be used specifically for females; the poet 劉半農 Liu Bannong (1891-1934) is credited with introducing this usage.
Could someone more advanced in Chinese tell me why "He isn't eating breakfast" is wrong?
That would be ”他不在吃早饭“—you need to have 在 for present progressive "is verbing"