The term came about in ancient history both to refer to the region in general (similar to the term "Greece") and to the people that lived there (similar to the term "Greeks"). One of the Sea Peoples that attacked Egypt and mentioned in contemporary records were the Peleset. The Egyptians defeated the Peleset, who eventually settled in the area of later Palestine/Israel (and leading to the Greek word for the region, Palaistine). Peleset gives rise to the Northwest Semitic Peleshet, which became Pelistim, and eventually was transliterated as Philistine in Greek.

By way of Greek, the term was used in Latin as well, where in Imperial Rome the region became part of the greater province of Syria-Palaestina through the merger of the provinces of Syria and Judaea. The province was later broken up, with Palaestina Prima, Palaestina Secunda, and Palaestina Salutoris retaining the name. After a brief period of Persian rule in the early 7th century, the Byzantines re-took it before losing it to the Arab Conquest. The Arabs, themselves inexperienced state-builders, opted to modify existing political systems to accommodate Islamic government, rather than strictly replace them. For that reason, they kept the name Filastin as a province in the greater region of Syria. The Ottomans kept the name in reference to the people and region, but reorganized the administration under Eyalets and later Vilayets (provinces), mostly named after Jerusalem specifically. This continued into the British Mandate of Palestine, after which it became Israel, and the citizens Israelis.

As for the second question, that's a thornier question to answer. Palestine has been under the control of foreign powers for millennia and so Palestinians have historically been a melding of these disparate influences. There has long been a "Palestinian people" of sorts, but the definition of who they really were was hard to answer. That is to say, it was rather straightforward throughout most of the region's history to determine if you were a Palestinian citizen: were you born there? Did you live there? If so, you were. But what that really meant or implied was much harder to answer. More recently (post-Israeli), even the first parts are in question. According to Rashid Khalidi in his book Palestinian Identity: The Construction of Modern National Consciousness:

... in Palestine many powerful and contradictory views of self and of history are conjoined. These may be religious, whether Jewish, Christian, or Muslim: or secular, as for example the focus on Masonic ritual on the Temple in Jerusalem; or they may be national or supranational, whether Arab or Jewish.

Later:

... [S]everal overlapping senses of identity have been operating in the way Palestinians have come to define themselves as a people, senses that have not necessarily been contradictory for Palestinians themselves, but can be misunderstood or misinterpreted by others. As Palesinian identity has evolved over time, its elements have varied, with some eventually disappearing and others newly emerging.

The first inklings of a true Palestinian identity started to come about in the late 19th to 20th centuries with the spread of nationalism. The disparate peoples of the Ottoman Empire began, before and after the partitioning, to explore concepts of ethnic identity on a local level, how they interrelated, and how they related to the Ottomans. After the partitioning, without the Ottoman Empire's cultural buoying effects, the different regions of the former empire were free to explore and establish their unique identities in an independent context. Palestine, because of its unique situation during and after the Mandate period, has lagged behind somewhat. Unable to define a nationality in regards to a geographic area like other countries (other than a relatively nebulous idea of "homeland"), they've historically had trouble identifying those unique Palestinian characteristics.

This may skirt the 20 year rule, but even to this day, there's tremendous debate amongst scholars about who the Palestinians actually are, and whether or not they're exclusively Arab, Muslim, or even have to be from Palestine/Israel.