On entering the West Bank, you will not have to show your passport and Israeli visa, but when leaving you usually will, either at the checkpoint on foot (Qalandia, Jalameh and Checkpoint 300) or to Israeli soldiers who board the bus (when leaving Bethlehem on bus 234 via Beit Jala).
Despite what you may be told on arrival in Israel, there is no law against tourists visiting the West Bank. However, Israeli citizens are forbidden from doing so under Israeli law.
There are no separate customs procedures when leaving the West Bank, although if you are in your own car, it may be searched and swabbed for explosives. Carrying bulky items through the checkpoints at Qalandia and Checkpoint 300 can be problematic, and if you don't think it will fit through an airport-style security scanner, it's probably best not to buy it.
There are no separate visa requirements for the West Bank.