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.)