Interesting takes on the issue of Israeliness for us immigrants can be found at Postcards From Israel and at This Normal Life. The bloggers confront the problems of being relative newcomers. I discovered that my view is somewhat more indirect, probably because I came to Israel at age 51 and did not bring expectations of becoming fully integrated, but neither was I old enough to come to Israel as The Holy Retirement Village. By now I am convinced that being an "outsider" is essential to me, and Israel is the best place for me to live out my outsiderhood.
I am fascinated by the conflict between individual and collective living here. Specialists in cross-cultural studies have ranked the USA as the most individualistic country, probably because the ideal there has always been a meritocracy and the national mythical hero is the Self-Made Man. By the time I left the USA in 1988, I felt the myth had morphed into worship of Selfishness and Greed, and I was disappointed in the direction things were taking.
Israel is only 57 years away from its collectivist beginnings, and even though it has waves of immigrants from everywhere, they are all supposed to be Jewish and therefore to consider themselves collectively part of The Jewish Nation. The Israeli mythical hero of the kibbutznik-paratrooper ignores the Arab, Beduoin, and Druse citizens of the State of Israel. I suspect they may think of themselves as second- or third-class citizens, but not as "outsiders," if for no other reason than the fact that they never lived "outside." As Israel moves towards capitalism, I expect the Corporation will very slowly replace the collective, and life here will become more comparmentalized and, yes, individualized. I guess there's many a slip 'twixt the cup and the lip, so plenty of things can happen to alter that course. It's so complicated!