What is Israel doing?

Now, I'm not the most informed person, but I'm very confused about what the hell Israel is trying to do in Lebanon. Their attacks in response to the kidnapping of their soldiers maked sense. But everything after isn't entirely comprehensible.

They attack the Beirut airport multiple times. The Beirut-Damascus highway is gone. They blockade the ports (which incidentally, makes it more difficult for Europeans and Americans to get out). Today there are reports that they hit a hospital. The Israeli government has alternately said they want to establish a 30 mile buffer, they just want to disarm Hizbollah, and that they're willing to talk cease fire.

As has been pointed out multiple times, disarming Hizbollah will be very difficult. They don't have arms depots, their weapons are spread throughout the homes of their supporters. Rockets are launched from backyards. To disarm the area would require a methodical house to house search of a very large area by a very large force. I'm not sure Israel is up for that (not militarily, but rather morale-wise).

The idea of a buffer zone reminds me too much of Stalin. And the only way they could realistically achieve that is occupation. Israel occupied Lebanon before. It didn't work out so well then. Heck, that’s what prompted the formation of Hizbollah in the first place. I know, ideally, the plan is for the Lebanese army to take over the area now controlled by Hizbollah, but the military is incredibly weak, I don't see that happening.

The cease fire is even less likely to happen without major international involvement. Hizbollah is not going to stop lobbing rockets into Israel, which means Israel can't realistically stop hitting targets.

This seems less and less likely to evolve into a regional war. Amazingly, most Arab countries are siding with the Israelis (though this is more out of fear of Iranian influence than a like for Israel). But I don't see how this can stop short of a UN Peacekeeping force (that can shoot back at people shooting them. This is important, but often overlooked in UN forces). The other option is another Israeli occupation, which will never end well.

Israel is rapidly running out of choices. If they don't get a peace established in the next few decades, there will be more Arabs than Jews in the country (those pesky birth rates get in the way). Another occupation isn't going to make the goal any more likely.