Some questions on Gaza

Coverage of Gaza has discussed the slide into violence since the ceasefire ended. Why is no-one asking why the ceasefire was allowed to lapse? Could Hamas’ decision to let the ceasefire end be because the citizens of Gaza were still living under a cruel blockade, with no signs of political progress in sight?
What does it even mean to be neutral in this conflict, to call for both sides to end the violence? Which decisions by which people would make this happen in Israel and in Gaza? What would be the relative political costs for these decisionmakers? Is there truly equivalence in this call for neutrality?
Does Hamas truly exercise military-style authority over all the rocket-firers in Gaza? What are the lines of control in the organisation? To what extent can either diplomacy with or attacks on Hamas affect the number of rockets fired into Israel? Clearly there is some relationship – but how close is it?
Coverage of Gaza, and the Israeli state, talks enthusiastically about Hamas. But what does that even mean? Is it the leaders of Hamas? The military leadership? The military wing? Everyone in a Hamas military uniform? Everyone who voted for Hamas? Everyone in Gaza?
Further to the above, a fundamental rationale for the Israeli offensive is that Hamas wants to exterminate Israel, to drive it into the sea, and is not rational in its desire for this goal. This claim is extremely common in the popular discourse, particularly in letters to the editor and comments to online news stories. If it is true, then it ultimately justifies any atrocity against Hamas. Why is this extremely common thread of argument absent from official comment, reporting and analysis? What does it mean to say that Hamas wants to exterminate Israel? If this is the rationale behind much international support for Israel, surely it is urgent that this claim is tested rather than left to stand unexamined?