I would like to know, strictly halachicly speaking, the position on wearing Rabeinu Tam’s Tefilin. furthermore if one decides to wear them, should they be worn separately (Rashi then Rabeinu Tam) because some scholars in Israel say that they should be worn together (one very small pair of Rashi and one very small of Rabeinu Tam, worn together)?
As in many cases, there is no one strict halachic position. The Shulchan Aruch itself presents a complex view of the matter. In Orach Chaim 34;2, the SA first says that a Y’rei Shamayim (G-d fearing man) should wear two pairs of tefillin, preferably together, but if that is difficult, than separately. In the next sentence the SA says that only a Jew with established and well known piety should wear two pairs, implying that this is not an obligation. As you have mentioned, there are varied customs in this matter. Many Chassidim put on tefillin of Rabbeinu Tam after tefillin of Rashi. Very few non Chassidic Ashkenazi Jews wear tefillin of Rabbeinu Tam. Some put them on in private. Among Sepahardim it has become more prevalent recently, and many wear two pairs together. I suggest you consult a Rabbi who knows you to discuss if it is appropriate for you.