You're right, a lot of people in the world do not understand the situation in Israel is much more complicated, the Israelis want to live in peace, the problem that the State of Palestine is conducted by the Hamas government and the Islamic Jihad, if done well it's like negotiate with al Qaeda or Hezbollah are organizations of terror did not want the existence of Israel, and worse than that Israel is trying to protect itself so well is sometimes harms innocent civilians but because Hamas shoots rockets at Israeli kindergartens, schools, and hiding the rockets basement home family living upstairs, so how Israel can protect its people and to give her some peace?? ... still not found a solution and it's really hard!
Seems like you covered just about all the media talking points there; but let me try and expand upon them a little further.
Yes, everyone wants peace. Hell, even Genghis Khan wanted peace - on his own terms; what others wanted was of no concern. Israel also wants peace of course - and also in its own terms. Let's not forget that the Palestinians have been under occupation for half a century, and more recently under cruel and sub-human conditions imposed on them by Israel. Remember, the official Israeli policy towards the population of the occupied territories is to keep them at a level consistent with an humanitarian crisis, while not pushing them over the edge completely - we know this thanks to Wikileaks.
Does this justify extremism among Palestinians? Not in my view; but it's the expected consequence of treating people like animals. It seems to me that the best way for Israel to "defend" itself is to stop beating Palestinians down, and stop the occupation, which in turn will lead to a decrease in the support for extremist tactics.
And then there is this attempt at whitewashing Israel's image by saying that it does everything in its power not to kill civilians - you think that Israel "sometimes harms innocents"? Have you even been following the news lately? I wouldn't call 80% civilian casualties in this conflict "sometimes". But what do you expect when you're bombing a densely populated area with warships, air strikes and artillery? And does Hamas' using their own population as shields justify Israel committing these atrocities, especially when it is the occupying power?