Is it correct to use "me too" and "I too"?

I'm a bit confused. Is it correct to use "me too" and "I too"? (Also with other pronouns.)

For example, if I want to say that Juan gives a present to Ana and I give a present to Ana:

Juan gives a present to Ana, [me/I] too.

Or if I want to say that Juan gives a present to Ana and Pedro gives a present to Ana:

Juan gives a present to Ana, [him/he] too.

When I write, I would not use any of these “[pronoun] too,” and I would write

Juan gives a present to Ana, and so do I.

Juan gives a present to Ana, and so does he.

When speaking informally, I may say

Juan gives a present to Ana, and me too.

but this is ambiguous because “me too” can mean both “I give a present to Ana, too” and “Juan gives a present to me, too.” I would never use “I too” in the informal context.

(By the way, all of these examples have a separate issue: using the simple present sounds strange because it implies Juan, Pedro or I give a present to Ana routinely.)

(Added: Kosmonaut pointed out in a comment that the simple present is used also when you are narrating the situation (thanks!). That is a more plausible context in which these sentences are used than people give a present routinely. See Kosmonaut’s comment to this answer.)

I thought to enlarge upon the first para of this superlative answer by quoting http://www.grammarphobia.com/blog/2010/08/me-too-i-too.html (I was the one who boled):

Q: I know someone who thinks he knows everything about English. This person says the most widely tolerated grammatical error is “Me too.” He insists it should always be “I too.” Is this true?

A: “Me” is a much misunderstood pronoun. Perhaps the most common grammatical error in English is using “I” where “me” would be correct.

For example, in a sentence like “He told John and I a story,” the pronoun should be “me,” not “I.”

In standard English, “me” is an object pronoun. “Me” is technically incorrect only when it’s being used as a nominative (or subject) pronoun – that is, when it’s the subject or implied subject of a sentence.

So “me” is impeccably correct in cases where it’s the implied object of an elliptical (or incomplete) sentence like “Me too.”

For example, if we say, “She invited us to the party,” and you respond, “Me too,” you’re using “me” correctly. “Me too” is an elliptical way of saying “[She invited] me too.” Here, “I too” would be incorrect. You’d never say “She invited I too.”

Or if we say to someone else, “Here’s a gift from us,” and you respond, “Me too,” then you’re using “me” correctly. “Me too” is an elliptical way of saying “[It’s from] me too.” Here, “I too” would be incorrect. You’d never say “It’s from I too.”

On the other hand, if we say, “We’re hungry,” and you respond, “I too,” you’re technically correct though unnaturally formal (more on that later).
In this case, “I too” is an elliptical way of saying “I [am hungry] too.”

There are other kinds of constructions in which the choice of “me” and “I” in short elliptical phrases may depend on whether a subject or an object is implied. We wrote a blog item about this last year.

So much for what’s technically correct and incorrect. The truth is that few people say “I too,” and for good reason. Even when it’s correct (and often it isn’t), it’s stiff and formal sounding.

As we’ve written before on the blog, the use of “Me too” for “I too” is an extremely common idiom and a natural development in English.

The reason is that English speakers generally choose “me” over “I” when a pronoun is the subject of an elliptical, verbless sentence, never mind what’s technically correct.

In a short reply without a verb, “I” seems unnaturally stiff to most people, including us. If it seems stiff to you too, use “Me too.”