How does Jesus have two wills in light of the rejection of Nestorianism? (Orthodox Trinitarian view)

Since I answered one of the original questions, I'll try answering here in a similar vein.

First:

Take the claim "Jesus can't be fully human without a human will;" why can't Jesus be fully human because He has a will as a person? As in, a will that is attributed to the person of Christ rather than to his individual human nature. I don't get how that wouldn't fulfill the "fully human" requirement. It seems that to say otherwise is just based on how we define what "human" is (which of course would be important).

Indeed that is an option. Some modern philosophers have decided that the council got it wrong and have decided to opt for monothelitism instead, while still accepting the hypostatic union. For example, William Lane Craig and JP Moreland in Philosophical Foundations for a Christian Worldview, and Garret J. DeWeese's essay One Person, Two Natures: Two Metaphysical Models of the Incarnation, where he quotes A. H. Strong:

The Logos did not take into union with himself an already developed human person, such as James, Peter, or John, but human nature before it had become personal or was capable of receving a name. Christ's human nature realized its personality only in union with the divine. At Jesus' conception the two natures vitally united to form one person with a single consciousness and will. Jesus' consciousness and will were ... always theanthropic -- an activity of the one personality which unites in itself the human and the divine.

In that same essay DeWeese admits that this model is explicity monotheletic, but suggests that the council may have had a different metaphysical understanding of "will" and so says: "it is not at all clear that this proposed contemporary model entails the view that was condemned in 681." (though this is a controversial claim and an active area of debate)

Now onto some of the questions:

If we can't attribute adjectives or actions to either individual nature, why can we attribute wills to the individual nature? How is that not separating the two natures that should be indivisible?

I agree when you said

it would be incorrect to say that "Jesus' human nature died on the cross, but his divine nature did not."

However we can say "Jesus died on the cross (in his human nature)" or "Jesus died on he cross (through his human nature)" In fact holding these things in tension is essential for Christian theology. "Jesus died" and "God is the only eternal, necessary being" are both true. The hypostatic union is an attempt to unravel that apparent contradiction.

But, if only His human nature slept while His divine nature was awake, then perhaps we could escape concluding that God slept? Except that the Bible seems to be denote the person of Jesus with actions or adjectives, rather than an individual nature.

Yes indeed. There are two ways to tackle this:

1. Divine Kenosis: whereby the Word voluntarily gave up divine attributes and emptied himself to assume human attributes

2. Divine Krypsis: whereby the Word voluntarily self-limits himself and only takes advantage of human attributes

   DeWeese again:

   the contemporary model explains the "self-emptying" as Christ's voluntary self-limitation to exercise his personhood through his human nature, gaining information about the world through the perceptual faculties of his human body, learning and storing memories through the instrumentality of his human brain, living a perfect human life by his perfect obedience and complete dependence on the Holy Spirit

(Radical kenosis has theological and philosophical problems that make it incompatible with Orthodox Christology)

So to come back to the question, we can attribute actions to an individual nature, but not merely a nature, but to a person through a nature.

Oliver Crisp (whose book I cite in the previous answer) offers one way to understand nature in this context:

(1) Human natures do not exist independently of human beings. (Human natures are concrete particulars) (2) Christ has a human nature in addition to a divine nature (3) The human nature of Christ exists because the Holy Spirit brings it into being. (4) This human nature of Christ does not exist independently of the theanthropic person of Christ.

Here it is the case that human natures do not exist independently of human persons, because human natures are concrete particulars. And this is the case for all human natures, not just the human nature of Christ. What is assumed at the Incarnation, according to this view, is a particular human nature, not merely human nature per se.

There are other ways of understanding "nature", but I think this view may help answer your questions:

How do do we know it is acceptable to attribute a property to one nature and not the other given that the two natures are inseparable? What does it mean for them to be inseparable if you can identify properties of each individual nature rather than the Person?

The two natures are joined in the one person of Christ. We can separate them theologically to try and understand them (to some degree), and we can nuance our theological language, but they can't be separated in reality. When the word became flesh the divine nature and the human nature were linked together, and that link will never be broken. When Christ rose again and ascended into heaven, he didn't lose his human nature.

And the human nature that Christ assumed never had (and never will have) an independent existence. Human natures cannot exist apart from human beings. And since the Word, the second persons of the trinity, had a divine nature before the Incarnation, the divine and human natures of Christ must coexist.

So they can't be separated, but we also have to be careful about confusing them. Because, as you said, properties of the human nature can't be applied to the divine, and vice-versa, properties of the divine nature can't be applied to the human.

But we can apply those properties to the one person who has both natures, in the careful, nuanced way Chalcedon is trying to get at.