Translation:This girl goes to school with her brother.
Here, ходит is used for 'going to school', meaning it's a "generic, multi-directional or non-specific" act of going.
If the intention had been to say she was going to the school along with her brother on a specific day (one occurrence, a defined direction), идти would have been used insteas (Она идёт ...)
According to here, http://www.russianlessons.net/grammar/pronouns.php, instrumental female reflexive pronoun is "Своей", why is "Своим" used here? Perhaps, unlike "Его" and "Её", "Своей" and "Своим" are determined via the noun's gender...
Его and её are possessive pronouns, hence the gender (его or её) is selected from the subject/person who is owning (possessing) something.
Своей / cвоим is a reflexive pronoun, not different words for for a male/female subject, but the same word used for all persons/numbers, just with declension to match the following object(s) being possessed.
It's no use, they won't add it. ’The school' isn't natural here. Children go to school to learn things. They go there for the purpose of studying It isn't the place of destination. Their parents can come(go) to the school (depends on context).
To put it another way: go to school means to attended a school. To go the/a school would imply they visited a school building. The same situation obtains with the words church, hospital, prison.