I am Lananaroux or Person A in this situation. Tyler James Young is correct as to what I meant by "Let me have it."
In my comment, I conceded that Southerners really do deal with quite a lot in terms of weather and surviving nature and the elements. However, I intended to imply that feeling like I'm hardier than Southerners during the winter months helps me feel strong enough to endure our long, cold winters. When I said "Let me have it" I meant: "Please just let me think that I am stronger for living here, where it is cold, than if I moved south, where it is warm, because whether or not it is true, the thought helps me through the winter."
I'd rather not be burdened with the truth of the matter, I would rather just have my feeling of superiority whether or not it is actually valid.
The verb "let" has two meanings... I was wondering, is it really so?
Yes, it's so (although, as I mentioned in my comment, those aren't even the only two ways the verb let can be used).
Cambridge labels these two uses of the word as to CAUSE and to ALLOW.
As for your second question:
What is the difference between:
4a) Let him call me when he is back.
and
4b) May he call me when he is back.
it's important to remember that some uses of a word fall in between what a dictionary might define; one particular use of a word may not clearly map to one definition or the other. For example, the word strange can mean "unusual or unexpected," and it can mean "not familiar or well-known". But it's not hard to imagine a context where the word could be used to mean "unexpected and unfamiliar" – that wouldn't be so strange at all.
Given a choice between your two sentences, I think you'd want to use the word let:
Let him call me when he is back.
That would sound rather natural in a conversation such as this one:
Is Dr. Davis there?
No, I'm sorry, he's out of the office. Would you like me to page him?
No, that's alright. This isn't an emergency. He can call me when he gets back.
That's probably how I would word it. "Let him call me when he is back" doesn't sound quite as natural, but I'd still find it to be acceptable English.
As for:
May he call me when he is back.
that should probably be avoided. The "May he..." sounds like the start of a curse or a benediction:
May he rot in hell forever.
May the Lord bless you and keep you safe.
I suspect you really mean:
He may call me when he is back.
But I still don't like that wording. That would be okay if you were disciplining a child:
Junior missed curfew; he'll be grounded for a week. He may call me when he is back.
Now, having said all of that, I want to address the crux of the matter. You're not really granting permission for someone to call you back, nor are you causing it. You're simply assuring the other person that "when he gets back" is an acceptable time to call. I think that's why I prefer can over let .
Notice how Cambridge defines the nuances of can: In addition to to be ABLE and to ALLOW – which are rather similar to the meanings of the verb let – there is also the meaning of to OFFER. I suspect that's what you're trying to do, to offer the individual the option to call upon his return.
I know this is a really long answer; I hope it is not excessive but instead clears things up for you. Indeed, the words can, let, and may have many overlapping meanings, and it's easy to see how they could be quite confusing in a context like this. I think you've asked a great question, one that would be interesting to natives and non-natives alike.
Best Answer
If you want to challenge someone to a competition, you could just say "Let's compete in <something>!" Or "Let's have a <something> contest!" More formally, swapping "Let's" out for "Let us" is possible, but that would be a little unusual. In either case, this is the sense of "I suggest that we […]".
If you want to ask someone to give you permission to compete, you would instead say "[Will you] let us compete, please?" (And presumably give some sort of reason why.) In this case, you cannot use the contracted form "let's" at all, because you're emphasizing the allowing sense of "let", and that is never what "let's" means.
"Let us" (in either sense) can also be changed to "let them" if you don't include yourself.
Neither "do" or "make" are really appropriate here at all, without major revisions to try to fit them in somewhere. And "let" without "us" or "them" immediately following won't work either.