Me too! Me neither! Yes we say them both. "I love chocolate." "Me too." "I don't like sweet things." "Me neither." (or neither do I) (I don't like it either) Me neither ... is probably considered the more informal way of saying it, and the other forms a little more formal.

Yes, it is correct to say 'Me neither'. It's an easier option than the other type of response which has to follow the grammar of the original sentence. 'Neither do I' is ONLY a response to a comment in the past simple: I don't understand it. Neither do I. Comments in other tenses need the auxiliary appropriate verb form: Neither did/can/could/have/had/will/would/shall/should/am/was I. And you do have say the correct one! Native speakers instinctively choose the correct form, but foreign learners might struggle to get it right - so for that reason 'Me too' or 'Me neither' may well be a lazy (but wise) choice.