Bonjour, 

I certainly do not know enough of French culture and language to discuss your explanation of French usage, below.  However, I disagree that in English, the difference between "may I" and "can I" is that "may I" is more formal.  More accurately, "can I" may be informal and in common usage, yes, but it is also often considered abrupt and somewhat impolite, and this is not only by grammar nerds (or oldies) such as I (or grammar nerds like me, to use more idiomatic speech, ha ha).

The grammatical distinctions are obvious, of course ("can" denoting capacity or ability, "may" is conditional, one is granted capacity).  One hears often, when someone is ordering in a cafe or bar, "can I get a... ".  To many English-speakers, this is grating and impolite.  I, for one, hope that "may I have...", does not go out of idiomatic usage (though perhaps that battle is lost and I am just raving...)

 

Love Lawless French, by the way,

Alice

"In the case of pouvoir, note that je peux becomes irregular in the inverted form: puis-je, in order to ease pronunciation.

This structure is very formal as we stated before, and is usually used to sound particularly polite or even a bit affected: the nearest English equivalent would be to use May I...?"