Please include me and john. Yes you could argue that the i form is idiomatic. There are two issues here. This is one of those instances where computers have outpaced our ability to adjust to new situations. Typically i would say simply please send me a copy of future emails or please include me in your mailing list if you insist on bending to.
Its such an informal context that you might come off a bit affected if you used i. For an example of proper i usage just flip the subject and object. Include takes an object pronoun in all its forms eg. In all those sentences i is a poor choice.
John and me a friendly reminder. The speaker is the subject of the sentence the one performing the action and so you use the subject version of the pronoun. Our field hockey group which included memyself went to the party. However it might sound more natural to say she is with john.
Jane and i are taking him. One is grammatical and the other is conventional. It has the sense which included in your examples eg. A reminder about john and i vs.
I would use john valencia and me if i were creating a caption in a photo. In terms of meaning or semantics myself means me personally so it sounds ok to use it but in terms of syntax or how words function in a sentence an object pronoun is required so me is the correct choice. For instance she is with me is correct. Also she is with john is correct.
Dont say between you and i and dont say the boss wants to talk with bob and i or thanks for meeting with john and i its me. The grammatical issue was covered quite adequately by gayathri vijayan. You use me when the speaker is the object the person being acted on. Including is a participle here from the verb include you might also say that it acts as a preposition.
I dont have as strong a feeling about to me and john or to john and me and when john and me are linked by an or instead of an and its even less of a compulsion to put john first. English language usage stack exchange is a question and answer site for linguists etymologists and serious english language enthusiasts.