To understand what Chalcedon did, and did not establish with regard to whether Jesus had one will or two wills, it might be a good idea to state the crux of the Creed agreed upon by the emperor and bishops on October 25, 451.
“In agreement, therefore, with the holy fathers we all unanimously
teach that we should confess that our Lord Jesus Christ is one and the
same Son; the same perfect in Godhead and the same perfect in manhood,
truly God and truly man, the same of a rational soul and body;
consubstantial with the Father in Godhead and the same consubstantial
with us in manhood; like as all things except sin; begotten of the
Father before all ages as regards his Godhead and in the last days the
same, for us and for our salvation, begotten of the Virgin Mary the
Theotokos as regards his manhood; one and the same Christ, Son, Lord,
only-begotten, made known in two natures without confusion, without
change, without division, without separation; the difference of the
natures being by no means removed because of the union but the
property of each nature being preserved and coalescing in one person
(prosopon) and one hypostasis, not parted or divided into two persons
but one and the same Son, only-begotten, divine Word, the Lord Jesus
Christ; as the prophets of old and Jesus Christ himself have taught us
about him, and the creed of our fathers has handed down.” [Gerald
Bray, Creeds, Councils & Christ, (Leicester, U.K., and Downers Grove,
III.: Inter-Varsity Press, 1984), p.162]
This Creed was careful to state the real humanity of Jesus Christ and his two natures, yet without separating or dividing those natures; that both are held together in one person. It clearly affirmed that the two natures of Christ must not be mingled or mixed, or thought to change. There is what has been called ‘the four fences of Chalcedon’ – “without confusion, without change, without division, without separation” – guarding, as it were, the belief in Christ’s two full and complete natures in one person. The definition does not explain anything. All it does is express and protect a mystery.
It did not prevent later arguments because Leo and Cyril, plus various orthodox and catholic interpreters, were still caught up in the old Greek belief in the divine impassibility. This enabled the Chalcedonian statements about Christ to be interpreted in increasingly Nestorian senses later on. Questions about whether Christ had two wills or one would not go away.
In the East were the dyophysites who believed in the two natures as radically distinct from one another while rejecting the idea of two persons after the union. The dyophysites did not leave the Great Church, their hero after Chalcedon being Theodoret of Cyprus, who had been vindicated at Chalcedon. Another hero of Eastern Orthodox theology was Maximus the Confessor (580-662). He defended dyothelitism – belief in two wills of Christ – and was martyred for his stance. Due to that, the sixth ecumenical council condemned monothelitism – belief in Jesus Christ as one integral person with two complete but inseparable natures, yet with only one, divine will – and required belief in two wills as orthodox doctrine.
Because of your sub-question 3 and the way an answer referred to Jesus apparently not knowing who had touched him for healing, I would divert into a different way of viewing this. In Luke 8:43ff, the woman who had hemorrhaged for years and wasted all her money on physicians, surreptitiously touched the hem of Jesus' garment as he passed by. Immediately, Jesus was aware that power had gone out of him. That was what elicited his question, "Who touched me?" His mind was utterly in harmony with his body because Christ himself had gone out (or, gone forth as healing into that woman's body). It was not that he was doing the work of healing, but that he went out as the healing. When he goes forth as healing, people get healed. Healing is not a matter of elimination but of having Christ as the one who heals. It wasn't a matter of something that left Christ but of someone who became that woman's healer, through her faith in that someone. Divine healing is not a matter of obtaining an object, but of God becoming our healing. I refer to this to show that this was not a matter of mind, with Christ, for his divine nature knew full well what had happened, just as his physical body 'told' him what had happened. The only matter to be resolved was for the person healed to speak up and so be identified, because there was a great throng crowding round Jesus. He knew someone had great faith in him as healer and he wasn't going to let that rare thing pass by unremarked upon. He wanted to commend that person, to strengthen faith, which is what happened. The person had to stand up and be counted! No more surreptitious faith - others needed to know the power of faith.
This example shows the need to search for how the mind of Christ is fully informed by his divine nature, even though his literal eyes had not yet alighted on the individual in question. Rather than use any divine aspect of his being to see the person (as he saw Nathaniel sitting under the fig-tree before Philip called him, Jn. 1:44-51), he took the opportunity to teach the crowd in general, and that shy woman in particular.
Apart from that point, much of this material has been culled from two chapters on Chalcedon in the book The Story of Christian Theology by Roger E. Olson (Apollos, 1999). I conclude with a direct quote from p247, re. the later fifth ecumenical council’s interpretation of the hypostatic union in 553:
“…while one may embark on the mental process of seeing in the reality the two natures of Christ, one must always return to the fundamental truth that he is one Person, the Logos made man, to whom belong both divine and human properties, and whose are all the actions and saying reported of him in Scripture, whether divine or human.” [Justo Gonzalez, History of Christian Thought, 2:97]
I hope this brief answer gives as simple as possible an over-view of a complex subject.