Re the joint ticket and cross voting system.
Inevitably there will be a point during the term of either President or Vice President when they will have to perform a duty involving contact with officials from Greece or Turkey. During these times they will be acting both as head of government and also as head of state, that is the dual role of the Cypriot presidential system.
The prestige and legitimacy of the two highest state officials will be enhanced if they are voted by both communities. Just think of the TC VP, let us say, accepting the credentials of the Greek ambassador to Cyprus. If he is voted solely by TCs and is president by simple rotation his stature is diminished. If, on the other hand, he is elected through a GC vote, his stature as head of the state of Cyprus is enhanced.
These may be communication matters, not directly affecting the substance, but often it is the style and not the game that counts.