"I teach you."
No. Doceo is for "I". (Ego) doceo. Here, in the sentence, an equivalent would be:
Ego te doceo.
"Te" is not the subject pronoun, it's the complement pronoun, like "me" in English.
Tu te doces, is grammatically possible, but I don't know if it makes sense. Maybe teaching yourself.
Tu me doces.
I think generally it takes two accusatives or an infinitive and an accusative. Lewis and Short lists a few other constructions but I didn't see one with a dative.
I can't think or find any examples of a dative being used after a quick search, however there maybe one somewhere.