母親在廚房裡做飯。 

父親在客廳裡看報紙。

哥哥在書房裡寫字。
I think "寫信"(writing a letter), "寫功課"(doing homework), "寫筆記"(writing notes) or "練字"(praticing to improve the handwriting) is better than "寫字". It's not wrong to put "寫字" actually.

姐姐在飯廳裡吃飯。

妹妹在外面唱歌。
弟弟在外面跳舞。
The word "外面" is acceptable here, you can also use other words like: "屋外", "門前"

Your Chinese is very good, all sentences are correct except “餐厅” is more native than “饭厅”
 
母親在廚房裡做飯。
父親在客廳裡看報。
哥哥在書房裡寫字。
姐姐在飯廳裡吃飯。(餐厅）
妹妹在外面唱歌。
弟弟在外面跳舞。