A quick interjection regarding the Christological question:
The post-Chalcedonian period settled quite decisively that Jesus did indeed have to grow in knowledge exactly like all humans do, including potty training and all the rest, that he could get ill and so on, and that he was generally subject to all the consequences of human finitude that we all are. He had to learn to read, learn to walk, learn not to poop in his pants, etc. He got colds. He yelled in pain when he hit his thumb with a hammer. He could have died of thirst, starvation, asphyxiation, exposure, dismemberment, or whatever if the situation arose. He had to double-check his measurements before cutting a board to make sure he didn’t screw it up. To say otherwise is to slide into monophysitism. So whether we feel comfortable with it or not, we Chalcedonians need to affirm that Jesus’ knowledge as a human being was as finite, as provisional, and as culturally conditioned as anyone’s.
[After further interaction in Was Jesus’ knowledge finite and limited? on the SVS forum]
Part of what makes this question so tricky is the assumption that human knowledge and divine knowledge are the same sort of thing.
This assumption puts us into the position of having to decide whether he has one mind1 with access to a single, shared pool of divine-human knowledge (which in practice would involve the drop of human knowledge being lost in the ocean of divine omniscience), or has two minds that operate in some sort of tandem, each accessing their own pool of knowledge. The first of these leans toward Apollinarianism, and the second towards Nestorianism.
We do better when we recognize that the word “know” does not have a univocal meaning in the sentences, “Bob knows the height of Mt. Everest,” and “God knows the height of Mt. Everest.” There is similarity between our knowing and God’s knowing, but there is always the ever greater dissimilarity, as the Fourth Lateran Council nicely phrased it. So when we talk about Jesus knowing things as a human and him knowing things as God, we are talking about analogous but nevertheless different actions. This is much like the analogy between how he created tables as a carpenter and created the cosmos as the Word.
Now, it is impossible for us to understand what divine knowing is like since, as humans, we can only know what human knowing is like. This in turn makes it impossible for us to imagine the way divine and human knowing relate to one another in him. But based on the observable evidence of Scripture, it seems not to involve negating or overwhelming the finitude of his human knowledge.