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.
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?
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.