Yes, Cnristians are certainly allowed to enter Al Asqua mosque on the Temple Mount in Jerusalem. Whether they pray or not while there is up to them. Chrisian and Muslim Pals currently cohabitate peeacfully in Israel and the territories (not to other primarily Muslim countries like Turkey and Morocco)- nothing indicates they would not continue to do so in a Palestinian state. You're poorly informed.
Syrian group says Nusra abducts its leader, in blow to U.S. plan (Reuters)
from the article: U.S. Jews and Israel versus Barack Obama