1. Most tourist sites - museums, etc - are open on Saturday. You have to check the websites. All National Parks are open. Most public transport stops from Friday afternoon until Saturday evening. Taxis are always available (the is no Uber). A lot of restaurants (non-kosher) are open. Most stores are closed (not all), but some reopen in the evening. Taxis are the only option from the airport on Saturday, but there is a shuttle service to both Jerusalem and Haifa.

2. Most tourists don't visit the area around Gaza, so unless there is a drastic deterioration in the situation you wouldn't be affected. You have to follow the news because nobody can predict what will happen next month. The same applies to the areas near Lebanon.

3. The answer depends on a lot of factors. How long is your trip? Will you be going to Jordan to see other sites (e.g Petra)? Are you aware of the logistics of traveling between the two countries?

4. There are plenty of attractions. You could easily spend a week in Jerusalem alone. There is a lot in Tel-Aviv and Haifa. There are dozens of interesting sites in the Galilee and the south. Again - how long is your trip? What are your interests? Will you rent a car? Etc, etc, etc.
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