There are some shades of meaning in the questions you've listed. But before I go through them: the most important thing in interpreting any of them will be context.
That said, I think the most useful way of splitting these up would be to say in what context you'd be most likely to hear them.
A. Shall we have a lesson on Monday?
As you correctly point out, you will probably only hear this in UK English, where it is a polite suggestion. An American equivalent might be:
How about we have a lesson in Monday?
or, slightly more aggressively but still polite:
Let's have a lesson on Monday!
Your "B" and "D" sentences will also probably be heard as a suggestion:
B. Can we have a lesson on Monday?
D. Could we have a lesson on Monday?
Here there's an implication that you're consulting the other person's schedule. You might hear:
Your next lesson is scheduled for Monday, but I have a dentist appointment, so I'm afraid I can't make it.
Well, then, can we have a lesson on Tuesday instead?
"Can" can also be used to ask if something is possible:
The Vice President will be visiting this week, and the whole campus is on lockdown, including the rehearsal rooms.
Oh, no! Can we still have a lesson on Monday, then?
while "could" implies a stronger desire (in American English, at least--in British English I don't think that's necessarily the case).
How would you like a lesson from the great Zanzini?
Ooh! Could you give me a lesson on Monday? I'd love that!
"Should" is interpreted in its usual meaning of "is it advisable to"/"is it a good idea to".
You need a lot more practice before the audition on Tuesday.
Well, then, should we have a lesson on Monday? Or should I rest my voice?
The last three:
E. Are we having a lesson on Monday?
F. Will we have a lesson on Monday?
G. Are we going to have a lesson on Monday?
are simple questions about whether an event will happen; you would probably hear them used in a case where there are a series of regularly scheduled lessons and you want to confirm that the next in the series is going to occur. For example:
This class will meet every Thursday from now until the end of December.
Are we going to have a lesson on November 27? That's Thanksgiving day.
or
I'm going out of town for the weekend, so I won't be able to grade your paper until at least Tuesday.
Okay. Are we still having a lesson on Monday?
or
I want you to spend all of your time between now and Thursday practicing your forehand.
Okay. Are we having a lesson on Monday, or should I just work on it on my own?
Any difference in which one of the last three you're more likely to hear will have more to do with regional variations than shades of meaning.
To my ear, the PEU examples are both colloquially natural and accurately described, while the Macmillan examples, although colloquial, are somewhat misleadingly described: things that are happening now might be better expressed as things that have just happened or have just been reported. But that's a minor criticism.
All these examples exhibit the same basic use of will to express habitual or characteristic action. In this sense (like all the modals in all their senses) this habitual will employs its tensed forms, and particularly the past-tense form, to express nuances which may have little or nothing to do with tense. Speaking very broadly,
Ordinary unstressed uses usually express simple habit—behavior repeated frequently over a long period of time—and tense has its usual significance of present or past reference:
When he's preoccupied with a problem he will often pace up and down for hours.
When I was a child we would always go to my grandmother's for Thanksgiving dinner.
When will or would is stressed, the sense is somewhat different: the habitual action is represented as perversely deliberate. And the tenses are employed somwhat differently, too. The present-tense form generally signifies that the subject currently makes a habit of the perverse behavior:
She will keep falling in love with the wrong people is rhetorically equivalent to She insists on falling in love with the wrong people.
But the past-tense form, although refers to a past action, does not express repeated action in the past but represents the subject's behavior on a single occasion as characteristically perverse:
You would tell Mary about the party ... Even though I asked you not to tell anybody without checking with me, you just can't control yourself around her.
A: He said it was all Jack's fault.
B: Well, he would say that, wouldn't he. Nothing is ever his fault, oh no!
Best Answer
Your understanding of the word "would" is wrong. When talking about future events the difference between "will" is referring to something that is going to happen in the real world, but "would" is talking about what would happen in a hypothetical situation. So for example, I could say
and that means that if I had more money (an imaginary situation) then I would go with you. Let's take another sentence.
This means that I have given up trying to get a day off, and I don't think it's possible. In a hypothetical where I do get off work, I would go with you, but in the real world I will not. However, if I say
that means that I don't know if I can get a day off or not. I'm still trying to convince my boss to let me have a day off, and I don't know if he will let me or not. If he does, I will go with you, but if he doesn't, I will not.
If I say "It will not happen", it does not mean it is unlikely. It means "In the future, this event is going to not happen." "It would not happen" is only accurate if you're talking about a hypothetical.
Now onto your last sentence:
It all depends on the context.
Let's say that I want to run a marathon. I sign up, and start training. The big day comes and I'm really excited. You want to discourage me so you say:
New scenario: Let's pretend that I've always wanted to run a marathon, but I am afraid of failing. I want to try, but I want to know that I can finish the race before I make an attempt. I come up to you and ask, "Do you think that I could run a marathon?" You want to discourage me so you say: