In reply to your questions:
1. The current Code of Canon of the Eastern Churches does not list the title Catholicos. The Codes formerly in force did equate the title of Catholicos with Major Archbishop, i.e an Archbishop with jurisdiction over metropolitans but without the title of patriarch. I assume the Syro Malankar Catholics use the title because it is the one traditionally used in that tradition.
2. All the above sites reference Mar Baselios as His Beatitude, the correct address for a major archbishop/patriarch not His Holiness.
3. See 1.
4. Untrue, the Catholic Church recognizes that other Sees are Apostolic, particularly Constantinople, Alexandria, Antioch, and Jerusalem.
5. Rome recognizes three Catholic and two Orthodox Patriarchs in Antioch, it can recognize two in India.
6. See 5.
Fr. Deacon Lance