Should I put myself last? "me and my friends" vs. "my friends and me" or "my friends and I"

The difference between "I and my friends" and "my friends and I" is purely a matter of courtesy - they are both grammatically correct. I would tend to stick to the latter though, as it a) is more commonplace, b) is considered more polite, c) seems to flow better.

Indeed, your example of 'incorrect' usage is incorrect solely in that the first sentence uses the accusative (objective) pronoun me, when you actually need the nominative (subjective) pronoun 'I'. The second sentence of that example is correct, since the pronoun needs to be in the accusative, as the object. You seem to understand this though; this is just to clarify.

There is a tendency in informal speech and writing to use object pronouns when conjoined with other nouns or pronouns, even if serving as the subject of a verb. You never hear this usage if the subject is not conjoined; that is, no native speaker would say “me went for some ice cream” but “me and my friends went for some ice cream” is actually quite a common usage produced by native speakers of all kinds.

This happens because what linguists would call the “unmarked” or standard, basic form for pronouns turns out to be the objective form—me, him, her, them, and the like. This is the form of the pronoun used when there is no verb:

• – Who wants a cookie?
  – Me.

• “Me and Mrs. Jones”

• “Me and Bobby McGee”
• “Me & Julio Down by the Schoolyard”

What happens is as pronouns in conjoined subjects get further and further from the verb, the impulse to change the default form into the subject form is weaker, and in informal contexts, is simply not followed. Now, in formal standard written English, subjects of verbs must be in subjective form, conjoined or not, leading to generations of schoolmarms correcting their students:

• Mrs. Jones and I
• Bobby McGee and I
• Julio and I Down by the Schoolyard

and the famous musical about an excruciatingly correct teacher of English:

• The King and I

The most fascinating thing of course is that generations of schoolmarms correcting students over the apparently perfectly natural use of objective pronouns in conjoined subjects has made everyone with even a little bit of formal education intensely anxious about using objective pronouns, causing them to hypercorrect and use subject pronouns even where object pronouns are correct: “just between you and I” is a commonly cited example.

Edit: As for the original poster’s actual question, there is nothing in the grammar of English per se about ordering of pronouns in conjoined noun phrases. It is a kind of grammatical etiquette to put yourself last, but there is no rule of grammar governing the order.