Okay, "shall" and "will" are used interchangeably often, but there's a slight difference. See my post herein to read about it: What is the difference between shall and may.
As for "shall" and "should", "should" is the past tense, imperfect, and past subjunctive form of "shall", so they have several overlapping meanings. The modal shall comes from Old English sculan meaning to owe. First-person present indicative of sculan was ic sceal, meaning I shall, whereas the past indicative and past subjunctive were both ic sceolde, meaning I should.
The old future tense rule has the following paradigm:
Simple Future: I shall, you will, he will, we shall, you will, they will
Emphatic Future: I will, you shall, he shall, we will, you shall, they
As for your examples above, "you shall not eat" is a command, i.e. "I order you not to eat", and is much more powerful than the simple "you should not eat", which is more of a suggestion such as "you ought not to eat." In your second usage of "shall", "you shall surely die" is far more powerful than "you will surely die." If you used "will" there, it would be more of a prediction that you're going to die whereas the usage of "shall" therein is prophetic: you are like a soothsayer; you can see into the future and so you are prophesying what shall happen. The usage of "shall" almost guarantees that the event will occur.
Here are some examples below using the formal English future-tense rule:
"I shall not be in school tomorrow." (a prediction)
"I will not be in school tomorrow." (you intend to skip school or you
intend on not coming)
"You will not eat that fruit." (a prediction)
"You shall not eat that fruit." (a command: "you had better not eat
"They will not pass." (a prediction)
"They shall not pass." (a command: "no way are they passing!")
"We shall die." (a prediction)
"We will die." (we intend to die; this is either suicide or a
guarantee of death)
"You will win the game!" (a prediction)
"You shall win the game!" (a prophecy or a guarantee of a future
Now, I want to reiterate that the examples above use the old future-tense rule whereon Fowler wrote his treatise in 1908. While some of the above usages still occur in English today although often inconsistently, many of them would be considered quite formal or outmoded. You should be aware of them, however, especially when you are reading the King James Bible or the English Standard Version, because this old rule is often followed in those versions of the Bible.