"He makes soup to a girl" makes no sense in English.
You could say "He makes soup for a girl" or "He gives soup to a girl" but you would never hear someone saying "He makes soup to a girl."
Need to get some natives or near-natives to proofread the English in this course. The errors are becoming a significant distraction.
It would, but it depends on the context. Without any extra context, 'for a girl' is better
The reason it is written as "fete" here is because that is the dative form of "fată." Since girl is an indirect object here, it must be in the dative case. If you look in dexonline.ro/ and click on the declinations tab when viewing a noun definition, you will see all of declensions for the noun by case, gender, and number.
It doesn't quite mean the same thing - "some girl" here has a negative implication about the girl, that she isn't approved of by the speaker or at least that him spending his time making soup for her isn't approved of! Stick with 'a girl' unless you are emphasising the smallness of his enterprise, perhaps in contrast to someone else's: "He makes soup for one girl, she makes soup for the whole street"