Israel is also known as the State of Israel or the Land of Israel. Palestine is also referred to as the State of Palestine or the Palestinian Territories.
Palestine has never been a colony of Israel. Israel took control of the West Bank and the Gaza Strip in 1967 after conquering these territories from Jordan and Egypt respectively, where they were annexed territories under military authority.
Palestine was known as the Palestinian Authority and was later known as the Palestinian Territories.
It depends on how you define Palestine. If you define it as the territories under the de facto control of the Palestinian Authority, it only borders Israel. If you consider only the territories of the West Bank, even those under de facto Israeli control, it borders both Israel and Jordan. If you consider all of the Palestinian Territories, also including the Hamas-controlled Gaza Strip, it borders Israel, Jordan, and Egypt. If you consider the former Mandate of Palestine that many Arabs use when they discuss Palestine replacing Israel, it would border Jordan, Egypt, Syria, and Lebanon (but no Israel since Israel would be gone).
Gaza, West Bank, and East Jerusalem are the Palestinian lands according to UN partitioning of lands between Israel and Palestine.
The British Mandate of Palestine covered all of current day Israel (including the occupied territories) and Jordan, as well as parts of Syria. Currently however, the term Palestine refers to the Occupied Territories where Israel has put the Palestinians previously occupying what is now Israel. The Israeli Defense Forces controls Palestine with an iron fist, though recently there has been efforts to bring the Palestinian Authority to power. This has not been successful as of yet because of many things, among them Israel's want for an Israeli Jerusalem without Palestinians and Israel's wants for better safety for their citizens (leading to much worse safety for Palestinians).
The Palestinian Territories consist of the West Bank and the Gaza Strip in Israel; the UN refers to them as the "Occupied Palestinian Territories' and are a priority of the UN High Commissioner for Human Rights.
It really depends on what you mean by 'Palestine'? There is no country today called Palestine, but in the past there were various territories referred to as Palestine, though it was never a well-defined independent country. Many people refer to Palestine as the territory controlled by Britain after the First World War, under what was called "The British Mandate of Palestine" (which was captured from Turkey during WWI). That territory today includes the countries of Israel and Jordan, as well as the West Bank and Gaza Strip territories. Others refer to Palestine only as the territories of the West Bank and the Gaza Strip. There are various other definitions, and to many it is a highly charged emotional and political subject. The population today of Israel is 7.5 million people. The Arab population of the West Bank and the Gaza Strip today is about 3.5 million (there is no accurate census). The population of Jordan is just over 6 million.
Bethlehem is in Palestine.
For most of the Twentieth Century (since 1948), the Jewish areas were called Israel and The Muslim areas were called "the Occupied Territories" or the "Palestinian Territories. The original plan was for the Palestinian territories to be called "Palestine" and be set up as in independent country, but the Palestinians rejected this plan. Prior to 1948, the entire area was called Palestine, though it was never under Palestinian rule. In 1948, the Palestinians almost had the chance for self-determination for the first time ever.
Not very many. The territories where Palestinians live tend to be almost entirely Muslim. But there absolutely are Buddhists in the neighboring state of Israel, and you should be able to find a Buddhist house of worship there.
Israel, in the area formerly known as Palestine until 1948.