All three are fine and accepted. "I do" is correct for the reason you mention. Just "I" is an abbreviated version, as you say. And "me" is correct since the "than" can take a noun and change its case. I rather like Wiktionary's summary:
Than functions as both conjunction and preposition; when it is used as a conjunction, it governs the nominative case, and when a preposition, the oblique case. To determine the case of a pronoun following "than", a writer can look to implied words and determine how they would relate to the pronoun.
Oops. This is the sound I meant, not the sound I said above. Is this the right sound? https://en.wikipedia.org/wiki/Voiceless_alveolo-palatal_sibilant