Using "will" and "won't" for habits
According to a grammar book (Advanced Grammar in Use,3rd Edition) we can use "will" in order to describe a particular habit in the present. e.g. Everyday he will wake up at 7 am and eat his breakfast. Is this the case for the negative form of "will"? Or is "won't" used to express annoyance, criticize a behavior and not to describe a habit? Is "won't" used in a correct way in the following sentences? "Every time I ask him how he's doing, he won't answer me" "Every time he drinks coffee, he won't have sugar with it."
Solution 1:
Is this the case for the negative form of "will"?
Will and won't are modal verbs, which the Oxford Dictionaries link defines as expressing necessity or probability. In the context of habits, it means they express what the person habitually does (will) or habitually avoids (won't). As such, if will is used to express a habit, then won't can be used with the same sentence construction to express the opposite habit.
Or is "won't" used to express annoyance, criticize a behavior and not to describe a habit? Is "won't" used in a correct way in the following sentences?
It could do so. Depending on the example, it could also simply describe a habit without any emotional content, or it could express annoyance at a habit. Arguably, your first negative example falls into this category. The sentence doesn't necessarily imply that not answering is habitual, but it also doesn't necessarily imply it isn't. Likewise, one could read emotion into the sentence, or depending on the broader context, it could simply be a statement of fact. In any case, the grammar of the sentence is correct.
Your last example does sound like a statement of habit, though "he won't" arguably expresses more a habit of choice / volition than a habit of action, which might be better expressed using "he doesn't". Again, your sentence is grammatically correct.
Solution 2:
It would be correct to say "I won't be going back to the store because of poor customer service.", but you could also say, "I won't be having any more automobile problems thanks to my great mechanic!", so it's more of a descriptor of "will not" in both contexts, not just annoyance or criticisms.