What is meant by "shall" in the examples below? Does it have to do with certainty or authority?
Don't worry. I shall be there to meet the train.
You/He shall die.
You shall go to the ball, Cinderella.
First it is necessary to understand the differences between will and shall. Each has a range of meanings, especially will. The usage of shall has dwindled down to a few prescribed usages while the usage of will is very broad. In many instances, will and shall have the same sort of meanings, for instance, both will and shall are used to predict future actions (Cambridge, Oxford, and Longman Dictionaries). Yet, this meaning and usage for shall is also coming to be classed as old-fashioned:
slightly old-fashioned (Cambridge Dictionary).
often sounds formal and old-fashioned (Oxford Dictionary).
Both will and shall are modal verbs and as such have only one form: they take no -ing or -ed or 3rd person -s suffixes. They also form questions without the standard use of do/does/did, as do all modals. Also as with all modals, will and shall can be negated as will not or won't and shall not or shan't (also old-fashioned--but fun!). Now, in what ways do will and shall differ from each other?
In speaking of what is to occur in the future, shall is used with 1st and 3rd person plural I/We shall. In contrast, will is used for future occurrences in all other instances of 2nd person and 3rd person You/He/She will.
In conversation, shall is used with I/We to offer suggestions or advice: Shall I write the message? I shall sing it, shall I? Let us think about it further, shall we? Let us bring the dog in from the rain, shall we? In contrast, will is used for giving orders: You will eat your broccoli at once. You will give the dog a bath directly.
In speaking about determination or certainty, shall is used to express certainty--that something must happen--or that you are determined that it should happen: I shall study hard for the exam. In contrast, will is used to show willingness to do a thing: I will proofread your essay, if you want. I will call later, if that is agreeable.
These are the three contemporary uses of shall, though it is used in some sayings such as "who shall remain nameless" and "As ye sow, so shall ye reap." On the other hand, will has five more meanings and uses ranging from making requests to stating expected or known truth (Oxford Dictionary).
Now to consider your examples. The speaker is predicting the future event of his being at the train. The speakers are using shall with You/He (2nd and 3rd person), which is an old-fashioned and extremely formal usage of the meaning that states certainty: You shall surely die, so shall we all. The same is true for Cinderella, with an added authoritarian certainty; after all, magical fairy godmothers are certain and authorities. However, in these cases, shall is really the only logical option because saying "You will die" implies a willingness on the dier's part, while "You will go, Cinderella" implies a command, which takes away from the fairy godmother's benevolence and ignores Cinderella's desperate tearful desire!