“me too”

In the cases above cited, I would say, "I do, too."

Of course, politeness would dictate that you not, arbitrarily and uninvited, include yourself in an activity, but this is not an English grammar rule.

(speedwell is snarky after staying up all night helping her fellow with his term paper on Polykleitos and the Pythagorean influences on classical Greek sculpture.)

"Me too" is perfectly grammatical, and you shouldn't pay attention to so-called "grammatical fanatics". It's similar to answering the question "who is it?" with "it's me." You wouldn't say, "it's I" unless you wanted to continue with "(it's I), the King of Monarch-land" or something like that.

"I, too" is not correct. "I, also" is also incorrect. For formal situations, "me too" might be a little casual. The following are more formal:

I do as well
I do too

(some people might say "me, also" is correct. It's marginal for me)

If you want to avoid the awkwardness of worrying about "me too", both your examples could be easily solved by saying "So do I".

a: "I want to go to the store."
b: "So do I."

a: "I have ten fingers."
b: "So do I."

As you suspected, "me too" is grammatically incorrect as "me" is in the accusative case. Technically, you are saying the persons wants to go to store and you, or has you as much as they have ten fingers. However, like all applications of language, culture and context can render such phrases as appropriate.