In such situations where there is not exact gender mentioned: Again - context matters! - This applies to all languages. To this example: There is only a distinction between a thing with a male or a female grammatical gender is being owned, but none for the owner's, so both are correct, (if context does not tell you more!).
You are right, -you must omit the article before possessive adjective when referring to family member in the singular.-
However there are some exceptions:
-if the family member is modified by an adjective or a suffix, then the article is required-
"lui è suo figlio" but "lui è il suo primo figlio"
"mia moglie è bella" but "la mia bella moglie"
"mia sorella" but "la mia sorellina" (my little sister)
Logically speaking, the first son is the oldest son, however "he is her oldest son" is an interpretation rather than a translation. DL asks for a translation and therefore requires the exact meanings of the words given, so since "primo" means "first" that is what DL wants in the answer
The gender of the adjective and preposition must match the gender of the noun with which they are associated. Thus, "il suo primo" is required for "figlio" and "la sua prima" is required for "figlia." In both cases the possession can be "his" or "hers." On cannot tell from the sentence. So the translation can be "he is his first son" or "he is her first son." "He is his first son" was accepted on 4/10/2017.
In the dropdown selection boxes "li tuo" is given as both her and his. The impression one gains from this structure is that the son's parent is masculine. To be strategically subjective, in English one often refers to, where gender is ambiguous, them/their/they to cover uncertainty.
suo = masculine sua = feminine