Poet12, welcome back to the Grammar Exchange. We haven't heard from you in a few months.
In your first example, you need to know that "be sick" has a dual meaning. It can be a progressive state:
I am sick. I've had a fever for the last three days. I'm sure I will be sick tomorrow and the next day as well.
or an action:
I'm so sorry. It appears that my cat was just sick on your carpet.
In the latter case, it is a euphemism for vomiting. (You probably already know all this, but I can't assume.)
In your example, given the choices you present, "am going to be" suggests the action meaning with a sense of immediacy. I would say that this is the best answer.
"Will be" can work in terms of the state of having illness, but if you already feel awful, this indicates that you are probably already sick, not that you "will be" at some future time (as opposed to: "You know, I'm starting to feel a little bit off. I'm thinking I will be sick by tomorrow."). The fact that I said "probably" shows that "may be" would definitely work here to indicate present possibility, but not simply "may". And besides, your title heading indicates that you are thinking specifically of verbs dealing with the future, not the present.
"Has" can't even work with the pronoun "I". Where did you get that from? It's not even worth talking about.
In your second example, "is retiring" is the best choice. Even though this is the present progressive
tense, the phrase "next month" shows that this hasn't happened yet, and the tense shows that the wheels have already been set in motion and the retirement is scheduled to happen. (Theoretically, we could also use the simple present "retires", but not "retire" with the subject "he".)
As for "will retire" and "may retire", these don't convey the same immediacy. If you see dark clouds gathering on the horizon, you say that a storm is coming. If you don't see such clouds, you can say "a storm will come" or "a storm may come", but this is more like conjecture.
Poet12, I hope this has been helpful. I must say, though, that the phrasing of your examples, along with the multiple-choice answers, suggests that these are problems from a textbook or an exam. I am here to help, and you can help me and my colleagues by letting us know the source and context of your questions. (In this case, your title "confusion of future forms" told me most of what I needed to know about context.) Also, I think it would benefit you as well as us if you would at least attempt to answer your own question and tell us why you think your answer is right (even though you're still not sure, or you wouldn't be asking the question). Among so many other things, this would help us to focus on what you actually need help with and not waste time explaining what you already know.
PS: An afterthought here:
I feel awful! I think I. .......sick.
In a real-life situation, you should say it just like that. Don't worry about inserting any words. The lady will know to step out of the way. Trust me.