Does it sound natural?
Normally, a girl’s family expect the man who has asked for their girl’s hand to have enough income to manage the costs of a usual daily life.

Sehen Sie eine Übersetzung

Very close!
Normally, a girl's family *expects* the man who has asked for their "daughter's* hand to have enough income to manage the costs *of typical daily life*.

"expects" rather than "expect". When conjugating verbs in English, a group (even though it contains many people) is treated as a single thing. For example...
My father, mother, and brother *expect* me to go.
My family *expects* me to go.

"Daughter" is more appropriate than "girl". Even though in farsi this is the same word, in English, daughter and girl mean different things. A girl is any young female. A daughter is parents' female offspring, and can be any age.

"Of typical daily life" rather than "of a usual daily life". "A" is not necessary here. To me, "typical" seems like a more natural word to use than "usual".