Like the Dictionary of Basic Japanese Grammar says:
ながら expresses an action that occurs concurrently or simultaneously with another action. The action expressed by Verbながら is always secondary to the action expressed in the main clause.
B: じゃ、コーヒーを飲みながら話しましょう。
Then, let's talk over a cup of coffee.
C: *じゃ、話しながらコーヒーを飲みましょう。
Then, let's drink a cup of coffee while talking.
In this scenario, the speaker and the listener have something important to talk about, and the speaker suggests to talk about it while drinking coffee. Therefore, the most important thing is talking, and the secondary thing is drinking. Note that the main verb of these sentences is the one at the end of the sentence, as it often is in Japanese.
Another question: do I always need a direct object? I wrote "私は本を読みながらかきます" and my teacher told me to add a direct object for かきます - is it really needed?
I think the prof is saying that they don't know what you are writing. For example, consider the following sentence:
私はかきます。
Doesn't this sentence feel a bit uncomfortable? We don't know what the person is writing, so it feels like something is missing from the sentence. If somebody just told me "I'm writing.", I would want to ask them "What are you writing?"
In your example sentence, 本を belongs to 読みながら, so the verb かきます is just like in my example... uncomfortably incomplete. Even if we suppose that the reader is expected to infer that the object of かきます is also 本, it's still strange to be reading and writing the same book simultaneously, so it feels like the sentence would need more background to make sense.
