Translation:He rarely eats with his family.
Huh, why is "He rarely eats with her family" wrong? And if it is wrong, how would one translate "He rarely eats with her family" ?
Grammatically, the possessive adjective (son/sa/ses) may be either "his" or "her". But what the grammar does not tell you is that it will be understood as referring to the subject of the sentence. Here are a few examples:
- Il est tombé de son cheval = he fell off his horse
- Il est tombé de son cheval à elle = he fell off her horse
- Elle est tombée de son cheval = she fell off her horse
- Elle est tombée de son cheval à lui = she fell off his horse
"He eats with his family rarely." <-- acceptable English variant, or no?
The translation for 'rarement' gives both 'rarely' and 'seldom' as possible translations.
But as far as I know, though they're interchangeable most of the time, 'seldom' is a little more frequent than 'rarely'
Can we make that distinction in French as well?
Not that I know of, and I'm not sure that we really make such a distinction in English either. The distinction you mention doesn't strike me as one people actually make in spontaneous speech/writing, i.e. it sounds like an academic distinction without much to support it. If you look up "seldom" the definition is "rarely" and if you look of "rarely" the definition is "seldom".
As far as I know, French does not have a synonym for "rarement".
Conventionally, in 3rd person singular, the object belongs to the subject.
Otherwise, it would be "sa famille à elle".