1. I will do my best.

2. I shall do my best.
(#1 is right, #2 is wrong? Am I right?)

3. I will/shall be 15 years old next year.
(We can use both 'will' and 'shall', right?)

4. I will die.
5. I shall die.
(Are both the same? Does #4 mean I intend to die? Does #5 mean I shall die {soon because I am very sick.})

In the US, "will" and "shall" seem to be pretty much interchangeable; "will" is used more often than "shall," however.

In British English, there's a distinction:
http://grammar.ccc.commnet.edu/grammar/auxiliary.htm#shall

1. "I will do my best" is the correct sentence. Both "will" and "shall" are modal verbs used to express future actions or possibilities. However, "will" is more commonly used in modern English, while "shall" is primarily used in formal contexts or for expressing strong determination or obligation.

2. In this case, "I will/shall be 15 years old next year," both "will" and "shall" are grammatically correct and interchangeable. You can use either one to express the future age you will turn next year.

3. In sentences 4 and 5, "I will die" and "I shall die," there is a subtle difference in meaning.

4. "I will die" simply expresses a future eventuality without any specific intention or indication of when it will happen. It is a statement about the inevitability of death.

5. "I shall die" can be interpreted as a statement indicating an intention or certainty about the future, which suggests that the person speaking is aware of an imminent or approaching death. It can also be seen as a more formal or poetic way of expressing the same idea as "I will die."

It's important to note that the context and tone of these sentences can greatly affect the interpretation. In general, "I will die" is a neutral statement about the future, while "I shall die" can have additional nuances depending on the situation.