Why is ăn needed here? Is it necessary when there is no other object?
Can the translation include "she cooks food and he sleeps"?
In some context, it can. But in this course, for the sentence you gave, it's corresponding translation has to be "Cô ấy nấu thức ăn và anh ấy ngủ." (She cooks food and he sleeps).
1) cook, boil
TRIGGERED haha j/k
In another example "tôi học nấu ăn", the English translation is " i learn how to cook". Why in this example is she cooks ??