grammaticality - 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.

Answer

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.)