There is no direct bus to the waterfront area from the airport. I think you would need to take the train to Tel Aviv and bus from there. There are a few buses and 4 train stations in Tel Aviv, all depending on where you're interested in going. I recommend going to the Port of Tel Aviv if you have no specific plans, and would take the train 1 stop to Tel Aviv Hagana and transfer to the 204 to Dizengoff/Ben Yehuda and walk from there. However, things will be quiet that early in the morning.
From that area, take the 480 to Jerusalem.
I wouldn't recommend walking from Jerusalem to Bethlehem. Better to take a bus. I think the Bethlehem buses are only from closer to the Old City and beyond, meaning you will need to take the light rail to Safra Square/City Hall OR take any of the 70 buses to Hebron Road and transfer there.
No issue entering/leaving Bethlehem multiple times at all.
A day-trip in Jerusalem + dead sea would e very difficult. For dead sea, your best bet is renting a car, and that can be a whole day of its own, so coupling with Jerusalem is too much IMO.
What days are you in Israel? Jerusalem is best not on Saturday as there is no public transportation and driving in some areas is difficult at best. Many restaurants and sites are also closed then. That could be a day to do the dead sea.
The only danger I see is the time issue! Between Tel Aviv, Jerusalem, Bethlehem, the Dead Sea, etc., you're talking about great places to visit!
BTW, regarding cabs: Israeli cabs cannot enter the Palestinian Area A and Palestinian cabs cannot enter Israeli Area C or cross the green line. Also Israeli car rentals cannot enter Area A and possibly also not Area B.