Not so much--being Twelver Shi'ites their views derive from their notion of the occulted/hidden Imam, and their support for the notion of Vilayat-e Faqih expounded by Khomeini. Unlike Sunni Salafist groups, they don't really call for the establishment of global Caliphate.
The "Israel thing" is big for Hizbullah, and I don't really know any major analyst (or, for that matter, member) of the organization who thinks otherwise. As for power, they want veto power in Lebanon so as to protect their interests, but actually refrain from maximizing their claim to either parliamentary or cabinet seats because it better serves them to not take on a major governing role. They certainly recognize that, given the sectarian complexity of the country, they will never be in a position to seize complete and direct power. This is why the brief May 2008 display of armed power, followed by Doha, the elections, and the 15-10-5 (with one of the 5 leaning towards the 10) cabinet formula works so well for them.
Very few Shi'ites in urban Lebanon think of themselves in tribal terms (hamula is likely to be more important), although you'll certainly find it in the Biqa. Hizbullah has traditionally recruited from among the poorest segment of the population precisely because it positioned itself as a movement of the downtrodden--a point that everyone recognizes--although today I would say that their support is probably relatively constant across all socioeconomic strata.
If anything, Hizbullah is probably Lebanon's only functioning meritocracy. This isn't to say that kinship linkages don't have some effects, but it is to say that they do not play a substantial role in policy development and implementation within the organization with regard to important matters.
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