"I and Jane" or "me and Jane"?

There are very strong indications that " x and me" or "me and x" are the actual grammatically-correct form in English (in concordance with French and with historical usage) and the horrible resonance of "I and x" is among them.

The problem with the "me" formulation is that it will never get by an editor or reader who has taken primary school English, except in a quotative construction relating non-standard speech. The influence of classically-educated grammarians runs deep, and the idea that subject pronouns must appear in subject positions has been so thoroughly drilled into people that "x and I" sounds right to most people, even though it is arguably wrong according to the natural grammar of the language. (If one substitutes "a group consisting of [members list]", "me" is the proper constituent.)

The upshot is that if you want to be taken seriously in a non-colloquial context, you pretty much have to use "x and I". "Me and x" or "x and me" will sound too colloquial or sloppy, and "I and x" simply sounds wrong to most folk.

There is nothing ungrammatical about saying I and Jane are going shopping. Similarly, there is nothing ungrammatical about saying He and you are going shopping.

However, it is not idiomatic to use those pronoun order constructions. It is convention in English to put the person being addressed first in pronoun subject order:

You and he are going shopping.

It is also convention to put the first person singular last:

She and I are going shopping.

You and I are going shopping.

You would rarely hear someone address a crowd like this:

Gentlemen and ladies, please be seated.

Again, it is style and convention to say ladies and gentlemen. If you want to go against style and convention, that is your prerogative. Understand that you may sound like a non-native speaker, uneducated, or simply eccentric.

To address you last point. Me and Jane are going shopping may sound better to your ear than I and Jane are going shopping, but that's not because it's grammatical. It's probably because it's colloquial (though strictly incorrect) to use the object pronoun me first in that construction. Possibly because then me stays far enough away from the verb to not sound quite so incorrect.

Me and Jane like tea. <-- ungrammatical