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What is the correct usage of these two words in terms of a sentence that references conditional future situations. Eg:
1.
a. If you do not leave tonight we can go out for ice cream.
or
b. If you will not leave tonight we can go out for ice cream.
2.
a. If he does not leave tonight we can go out for ice cream.
or
b. If he will not leave tonight we can go out for ice cream.
1a and 2a sound more correct/feels more natural to me, but is one more acceptable/correct than the other?
Will is ordinarily not used in if clauses to signify mere futurity. It is used only in a handful of situations:
If it is employed in the sense of be willing (to VERB)—or, in the negative, refuse (to VERB):
If you'll get beer, I'll make hamburgers. If you won't get beer, I'm not going to make hamburgers.
If it is employed emphatically with the sense of insist on VERBing—again, the negative has the sense of refusal:
If you will ask silly questions you must expect silly answers.
If you won't follow instructions you must expect failure.
If it is employed to signify that you accept a prior prediction as fact—here the negative implies a negative prediction:
If (as you say) your mother will be here tomorrow, we'd better get clean sheets for the spare bedroom.
If (as you say) your mother won't be here tomorrow, there's not much point in getting clean sheets for the spare bedroom.
Consequently, unless what you're speaking about involves one of these situations, you want to use do:
If {you don't / he doesn't} leave tonight we can go out for ice cream.
—But as the commenters point out, we're more likely to use BE going (to) here than bare do.
1a and 2a are standard first conditional sentences. In these cases, it seems that it is necessary that someone remains at home (to watch the children, wait for the cable guy, answer the phone, etc.). Under that condition, we (the rest of the family/group) are free to go out and have delicious ice cream. We have passed our responsibilities to a proxy and now it's time to enjoy ourselves.
1b and 2b I think are also standard English, but with a different meaning. The use of "will not" indicates a refusal on the part of someone. In these cases, I hear an emphasis on the pronouns: "If you will not leave tonight we can go out for ice cream." and "If he will not leave tonight we can go out for ice cream." Here, the sense is that despite the childish insistence of "you" and "he" that cake is the only dessert worth leaving the house for, the rest of us are still capable of enjoying our delicious ice cream.
"Will" has the connotation of a choice to perform the action. "Do" more simply centers around whether the action happens.
Taking the prefatory clause "If you will not leave tonight", the connotation is that the subject of this clause can decide whether he "does" or "does not" leave. The clause "If you do not leave tonight" is similar overall but has the connotation that whether the subject wants to leave or not may not be the final say in the matter.
Overall, the difference to American English speakers is typically that "will" is used to emphasize that it's the person's own choice and nobody else's whether the action is performed, and thus the hidden meaning is usually "What happens next is all on you", while "do" is more neutral, implying no choice at hand, so the meaning behind it can more easily be "the future depends on this condition but I know it may not be completely up to you".
For this reason, "do" is heard much more often in American English because it's more polite-sounding, implying that the present or future situation is not necessarily the result of the subject's choice in the matter.
