Typically (though maybe not strictly) you can only say "I too" in a complete sentence. For example. "I too see the dog." or the famous, "I too like to live dangerously." http://knowyourmeme.com/memes/i-too-like-to-live-dangerously
If you are giving a simple response, however, you would say "Me too." This is a peculiarity of English that I never understood, but nominative (subject) pronouns cannot exist without a verb in modern English.
Incorrect → Correct
I too → Me too
He/She too → Him/Her too
We too → Us too
They too → Them too
I wouldn't say that it's incorrect to say I too, though. Originally, I too was the correct form of an implied sentence by omission, but because of the grammaticalisation of the objective forms as acceptable pronouns for certain constructions, the objective forms like me too and it's me became grammatical. That's not to say that using the subjective forms are no longer grammatical, just that they now have a pedantic register when not in a complete sentence. So it should be
Pedantic or formal register/casual
I guess I'm super late on this, but "I too" and "me too" are both correct, it just depends on what the rest of the sentence is. "Me too" could be short for "The same goes for me too," or you could say "I too" as in "I too agree with this." Most people don't seem to understand the practical differences between "me" and "I", though. My favorite example of an incorrect use of the word "me" is "He can run as far as me." In other words, "He can run the distance between me and him."
I'm a native speaker, and I consider everything you've said above to be incorrect. Whoever told you that nominative pronouns can't exist without a verb is as wrong as one can be. I know people use the accusative to such an extent that it's become correct English but that doesn't negate the original correct forms, which I and many other speakers of good English would always use.
"nominative (subject) pronouns cannot exist without a verb in modern English"
Sure they can. A perfect example of this is "I too", which plenty of native speakers see as perfectly correct. A lot of so called rules of English grammar aren't that at all, but rather rules of style. There might be some style guide that prohibits "I too", but that doesn't make it in any way ungrammatical.
It should. Whether you say "me" or "I" depends on the context. A lot of people use accusative forms when they mean the nominative. It doesn't make it correct. The main point here is that Duolingo is not accepting correct English.
It depends on the context if "Me too" or "I too" is correct. https://www.grammarphobia.com/blog/2010/08/me-too-i-too.html Therefore duo should accept "I too"!