He may do it, he might do it

You just showed me that I never used “Might”.

Or that “might” makes right?

In seriousness though, the reason I asked whether those examples all sound wrong to you is that I wondered whether it was a British usage.

It sounds “right” to me but a bit formal and archaic. It’s a line I might expect to read in a Sherlock Holmes novel.

The worst is:

“Do you know anything about John’s plans with respect to the trip?”
“Well, I think his mother said he may not go.”

Does this mean his mother is not allowing him to go, or is she expressing the possibility that he might not go?

That’s a good question. “May” does suggest permissive while “might” is decision. That’s the same impression I got with:

“Might I have a moment of your time?” doesn’t sound right to me. This question is asking permission so I thought “may” fits.

If it is verbal, you can extract meaning from inflection.

Well, I think his mother said he may not go,” emphasis on the negative, suggests lack of permission.

Well, I think his mother said he m a y not go,” with a rising tone, drawing out or sometimes just emphasis on “may” suggests decision.

Those, of course, cannot be expressed in writing, where the meaning is, AFAICT, completely ambiguous.