These are deep questions that are difficult to answer, but I'll give it a shot. Most of what I'm going to say is derived from Oliver D. Crisp's book Divinity and Humanity.
How far is this model to be extended? I have heard it said that not only does Christ have two wills, but also two minds. Is this an accurate interpretation of Chalcedonian theology?
First this particular issue flows out of the later sixth ecumenical council, so technically its possible to accept Chalcedon without also accepting dyothelitism (two wills), but its generally considered that Chalcedon entails the findings of the later council.
If you read the proceedings of the Council of Constantinople, the reasoning flows from a basic principle exposited by Gregory of Nazianzus:
For that which He has not assumed He has not healed; but that which is united to His Godhead is also saved.
So the model has to be extended far enough to cover everything that makes a human being a human being. If we take mind to consist of things like "will", "desire", "perception", "reason", etc. Then I think the answer is yes, dyothelitism entails that Christ has two minds, a human mind and a divine mind. For the divine mind is wholly different from a human mind:
“For my thoughts are not your thoughts, neither are your ways my ways,” declares the Lord. “As the heavens are higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts.
Similarly the council takes up several passages from scripture:
our Lord Jesus Christ himself, who is true and perfect God, and true and perfect man, in his holy Gospels shows forth in some instances human things, in others, divine, and still in others both together [...] Thus as man he prays to the Father to take away the cup of suffering, because in him our human nature was complete, sin only excepted, "Father, if it be possible, let this cup pass from me; nevertheless not as I will, but as you will". And in another passage: "Not my will", but yours be done.
"Not my will", pertain to his humanity; through which also he is said, according to the teaching of Blessed Paul the Apostle of the Gentiles, to have become obedient unto death, even the death of the Cross. Wherefore also it is taught us that he was obedient to his parents, which must piously be understood to refer to his voluntary obedience, not according to his divinity (by which he governs all things), but according to his humanity, by which he spontaneously submitted himself to his parents.
It keeps going in this fashion. But how can one person have two wills?
Here are two theological concepts which may help:
Perichoresis (mutual indwelling or interpenetration)
This doctrine is often used in discussions of the Trinity about how the 3 persons of the Trinity interrelate to one another, but it has also been used to describe how the human and divine natures of Christ relate.
Christ's humanity is in an intimate, perichoretic relation with the Word, which [...] is of a dgree of intimacy not enjoyed by other creatures
In Crisp's view the interpenetration is the result of "divine omnipresence" and that "penetration is asymmetrical: the relation originates in the divine and moves in the direction of the human nature only."
Because omnipresence means that the divine nature penetrates all of creation, Crisp sees the nature-perichoresis of Christ as "a difference of degree, rather than of kind". He offers this analogy:
A sword could be said, in a loose and non-philosophical sense, to be 'penetrated' by the heat of the blacksmith's furnace as he forges the blade [...] Presumably, if I were to place another sword in the furnace for a moment, it too would be 'penetrated' by the heat of the furnace and would become warm. But it would not be as hot as the first sword, which is being forged, and is a lot hotter than my own sword. The difference is one of quantity of heat, not quality of heat.
Though he's also careful to note that the hypostatic union is more than just nature-perichoresis. At any rate, this suggestion can make some sense out of Chalcedon's:
two natures, inconfusedly, unchangeably, indivisibly, inseparably; the distinction of natures being by no means taken away by the union, but rather the property of each nature being preserved, and concurring in one Person and one Subsistence
(As an aside, Perichoresis has an analogy to the eastern doctrine of theosis and the union we desire with God through sanctification... certainly not to the same degree, for Christ is unique, but it's definitely interesting to ponder)
Divine Krypsis (self-concealment)
In a discussion on divine knosis (self-emptying), Crisp suggests divine kryspis (self-concealment) as a more orthodox way to think about what's going on in the hypostatic union. The divine nature, though having essential attributes like omniscience and omnipresence, chooses not to use them during the period of his earthly ministry.
This krypsis Christology preserves what is required by a full-blooded Chalcedonianism: the Word does not relinquish or abdicate any of his divine attributes for the period of the earthly Incarnation. And it also makes sense of the notion of self-emptying that is alluded to in the New Testament: the human nature taken on by the Word in the Incarnation is limited and does not have access to those divine properties which the Word exercises
Hope that Helps.