If the recipient has your phone number and name in their phone contacts, your name should come up as the SMS sender when you send them an SMS.
Are you sure it says "ME"? This isn't normal, as far as I am aware. If they don't have your name and number in their contacts, your phone number should just be listed as the sender. Are you blocking Caller ID, by any chance?
I don't believe Android uses header information for SMS. The most you can do, is add a signature (usually can be found in your SMS app's settings). But this is really just a part of the SMS text, so it uses up part of your 160 characters.