No, "pomoj" is actually correct here. But your question is great. "Kiom da pomoj" is the direct object, so it's very reasonable to wonder about this. However, note that "da pomoj" is a prepositional phrase inside of the direct object "kiom da pomoj", and the object of the preposition "da" is always nominative.
Oh yes. I didn't think about the preposition there... But to really stretch this, take this sentence. "Kiom da virinoj vi havas?" It might be a bit confusing at first whether this is "How many ladies have you" or "How many ladies do you have?". I know. Kind of a silly sentence... I guess if it was the first, "vi" would have to be "vin". which means that the only reason you know which is the object and which is the subject is because the one that CAN be marked isn't.
Well, no. In English, the object of the verb "have" is still "women," and the subject is "you," and Esperanto is the same. You're getting confused because the question form reverses the order: "We have five women. How many women do we have?" Versus "She has us. How many of us does she have?" One automatically uses the object form, "us," not the subject form, "we."