Hey,
I did a summer program at the University of Haifa and would not recommend it. The other students were great but the University isn't particularly well-organized and the campus is very far away from the city proper. I've heard good things from people who studied at the Hebrew University of Jerusalem.
I also spent a semester in Tel Aviv working for a joint Israeli-Palestinian NGO. There are definitely passport issues. You would likely have to get a new copy; you won't be allowed to enter Lebanon with evidence of a visit to Israel in your passport. If you're American, embassies will even issue you a dual copy for this purpose. I don't know what the policy is for other governments. But it shouldn't be very difficult to get a new/dual copy after you leave Israel.
However, it's my understanding that beyond passport problems there shouldn't be any issues working elsewhere in the Middle East afterwards. I spent six months in Israel and was then hired by the UN Relief and Works Agency in Jordan without any problem. The time I spent in Israel working on Palestinian issues was seen as a positive. If you spend time volunteering or working for an organization that is seen as very pro-Israeli or anti-Palestinian, then that could definitely prevent you from being hired. But simply studying or volunteering in Israel shouldn't count against you. Many of the people I've met here who work on Middle East issues have spent some amount of time in Israel.
As far as Israeli visa issues: I got kicked out of Israel on a visa issue, which is how I ended up in Jordan. The Israeli government is very, very strict and it is indeed very difficult to stay, even if you marry an Israeli or convert. I spent like a month going down to the Israeli Ministry of the Interior every morning at 6:30 AM to wait in line, just to be told I was missing paperwork that had never been previously mentioned. It was exhausting. Not that this should prevent you from studying in Israel, just don't expect to stay after your initial student visa expires. Though if you apply for an extension the case can be pending for several months. Mine was pending for three and during that time you can legally stay in Israel. I have no idea what it's like in Lebanon.
From the language classes I took at U of Haifa, it seems people with a background in Arabic pick up Hebrew very quickly and vice-versa. You could probably learn a good amount in a semester if you already have some Arabic.
Hope that helps!