To address your initial query: Yes, you may. You may say to your child, "You shall do ...." You have the power to influence and control events and circumstances because you are a parent and therefore may say "You shall." Remember that in past eras, even into the early twentieth century, it was quite common for educated people to regularly say, "You shall" even though in contemporary English usage "shall" is suggested as restricted to "I shall" and "We shall." Usage of "you shall" in contemporary English currently depends upon the circumstances and the speaker: the speaker of "You shall" has some power to influence and control circumstances and the outcome of events. Remember though, time changes usage of vocabulary since in earlier eras, "they shall," "you shall," "he shall," and "she shall" were as common as "they will," “she will,” “he will,” and “you will,” are today.
One problem you're having in sorting this out is that when trying to understand the meaning of one modal verb, for instance "shall," you substitute another modal verb in the paraphrase, for example, as in your statements: “‘You shall …’ means kind of ‘something ... will...’ and ‘must happen’ is like ‘You shall ....’” This confuses the attempt to understand because each modal has a specialized meaning: Modals cannot be substituted for each other in paraphrases that are intended to facilitate learning the specialized meaning of one particular modal verb. [Bear in mind that sometimes language is used loosely, without due attention to the meanings of modals and that, therefore, some people advocate that "they all mean same thing and can be used interchangeably." While this may be true for some people in everyday, commonplace usage, such a "one size fits all" attitude is not applicable at this moment to you and your questions.]
Substituting one modal for another confuses the issue of understanding them. Let’s consider “must.” Must indicates necessity and obligation. By way of illustration, South African public buses carry signs that say "You must not smoke." The government has imposed the necessity of not smoking. The public riding on the bus has the obligation to comply and not smoke. How does this compare to "shall"? Shall indicates that the speaker has the power to influence and control events. Government exerted it's authority and says, "None shall smoke on buses." Then government says, "I shall impose a necessity. Make a sign that tells the public 'You must not smoke.'" The government further says, "The public now have an obligation. They must not smoke." How does "will" become involved in this story? Will indicates freedom to decide, choose, and act according to one's own volition. The public exerts it right to to choose what it will do. The story continues: "What will he decide to do? He will choose to honor his obligation and not smoke since government has imposed a necessity. "
You can see each modal has a very particular meaning and particular implications. You can also see, that while trying to comprehend the particulars of their meaning and usage, they can not be substituted one for the other because none means precisely the same thing: "must" indicates that a person has imposed a necessity that another person has an obligation to fulfill; "shall" indicates the speaker has the power to influence or control events or circumstances; "will" indicates a person has the power of their own volition to decide, choose, and act.