#1 is incorrect. We cannot use too with don't.
#2 might technically be correct, but it sounds a bit formal somehow—or maybe just weird. I don't know anyone who speaks this way in any situation. And changing the sequence absolutely makes it incorrect: I don't like tea also does not work at all. So I'm hesitant to call your version correct.
#3 is the best, most natural way to communicate the idea. It will work in formal and casual situations alike.
#4 is incorrect in most dialects, including the standard dialect. Matching neither with don't creates an inappropriate double negative.
#5 is incorrect for the same reason #1 is incorrect. In its place you can substitute Me either or Me neither. Both of these are quite casual.