Is this correct use of a semicolon:  ''It is hot today; I will eat ice cream.''?

Yes, and I find no issue with using "will".  I think I'll eat some ice cream; it is so very hot today.  Semi-colon use is a tough one, because it is typically only employed when the two clauses are distinct yet strongly linked.  In this example, the one clause is used to answer the "why" for the other clause.  No one asked why, but the declaration does make that a likely question so giving the answer as a part of the declaration is not unexpected nor unusual.

Yes. But it's an incorrect use of "will". "Will" show a promise or a commitment. "going to" shows a plan. "Will" could be used in specific circumstances.

"You shouldn't eat ice cream- it's bad for you."

"It's hot- I WILL eat ice cream, no matter what you say!"

But normally- "It's hot today, so I'm going to eat some ice cream." or "It's hot today; I'm going to eat some ice cream."

Yes, I know your teacher told you that "will" is the future tense in English. It's not. It has a specific meaning. The usual way we refer to future time is the present progressive.