Your explanations are very clear and show an accurate understanding of the various meanings and usage of the word shall. When thinking about these types of sentences though, I would caution you about "making up a story" about the sentence -- it could mislead your understanding if you are trying to learn how to accurately translate. In general, in English "shall" can be used where "will" could be used. As you noted in #2 above, there is a connotation of a more emphatic intent.
The other way shall is used in formal English is with 1st person questions. (3rd person questions use "will.") Here are a couple of examples:
1. Shall I make dinner for the family?
2. Shall we go out to eat at the local diner?
3. Will he make dinner for the family?
4. Will they go out to eat at the local diner?