The 1922 Census found a population that was 78% Muslim & 11% Jewish. By 1945, it was 60% Muslim & 31% Jewish. I'm not sure how good a job they did estimating the Jewish population given the high degree of illegal immigration.

The Arab population would have likely considered themselves "Arabs". The term Palestinian to refer to the Arab population was not used as it is today. The 1964 PLO Charter used the term Palestinian Arab. A lot has been written about the extent to which a Palestinian Arab nation existed. Rashid Khalidi argues that it's origins lie in the early 20th century though it's a disputed topic.

As for whether the land was stolen, I'll make a couple of points. It's easy to see how massive immigration caused tensions. Imagine if 100,000,000 Chinese people moved to the United States. Second, the issue of "stolen" land comes up in the aftermath of the 1948 Arab-Israeli war. This is a lot more controversial than the issue of Palestinian national identity. There were a whole lot of Arab refugees created that were refused integration in the bordering countries. Egypt occupied the Gaza Strip & Jordan occupied & annexed the West Bank, so no separate Arab state was created in the former British Mandate. A lot of the land abandoned by the the refugees were taken up by the state. To many that is theft, but it's kind of the story of history.

As far as I know there was no Palestinian nation as the region had been ruled by the Ottomans for hundreds of years at that point.

Real quick, I'd like to distinguish between a "nation" and a "state". A nation, in international relations, is a group of people who share a "national identity". It's a way of classifying a group who feels they have a common identity, to make the description brief. A state is a nation-state: sometimes it overlaps with a nation (ie. it is formed for a particular nation), sometimes it does not.

Now, the Palestinian nation definitely exists and existed prior to the 1948 war, though it wasn't very prominent among the people living in the area we call "Palestine" until past the 1920s. There was never a "state" that filled the area we call Palestine as independent and sovereign: at best, parts of Palestine were considered an administrative unit and geographical area of reference under the Ottoman Empire. People certainly felt they lived in "Palestine", but knew they were citizens of the Ottoman state: Palestine was an area, not a state itself.

Which nationality did the people of Jerusalem, Yafo, Bersheba consider themselves to be at the time of WW1 and after? Ottomans? Palestinians?

This is a very interesting question, and one which many have devoted time to studying. The consensus among Palestinians is that they had a feeling of multiple identities. Before the fall of the Ottoman Empire, people felt themselves to be primarily Ottomans, and identified as such. After WWI, when the British assumed the Mandate (and just in general after the war) there was a brief time during which the majority of people in the area identified merely as "Arabs", and were hoping for a pan-Arab state (which was the goal wanted by Faysal Husayn, the person who led the Arab Revolt against the Ottomans during WWI, worked with the British on it, and was leading Syria). Pan-Arabism was short-lived, however, in terms of being how Palestinians identified themselves. While newspapers and people in the area called for joining with Syria to make "Greater Syria", and Husayn's first Syrian Congress pronounced him king and attempted to say that Palestine was within his borders, Husayn was not destined to stay king for long. The British had promised Syria to the French in the San Remo Conference, and the costs of protecting him were far too high. They withdrew in 1920, leaving the French to send Husayn an ultimatum as a pretense for removing him, which he promptly agreed to follow. The French, dismayed but not discouraged, marched on Syria anyways and quickly removed Faysal, who was given back to the British and installed as king in Iraq, to protect British interests there.

After the fall of Husayn, there was a dramatic drop in people calling for a pan-Arab state, and the ideology of pan-Arabism faded fairly quickly into the background. Instead, papers like Filastin (Palestine) began calling once more for an independent state in Palestine, and the people began to identify in numerous ways: as Arabs, as Palestinians, and as Muslims. The goal, though, grew to mainly be for independent Palestine. The Muslims and Christians who espoused this goal, most of whom were Ottoman subjects, still identified in other ways as well, but the "nationality" began to be considered "Palestinian", since "Ottoman" no longer existed, leaving a vacuum to be filled by other identifying factors.

Are there historic arguments to back up the claim that the Israeli immigrants "stole" the country from someone, or was the area mostly uninhibited and both Palestinians and Israelis aimed to found a nation there?

There were, indeed, many people living in the area before the Zionist groups began to arrive. They frequently clashed leading up to the establishment of the Israeli state, as a result of the differing aims. The Arabs of the area felt they had been promised an independent state by the British, who were administering the Mandate. At the same time, the Zionists felt they had been promised a state in Palestine. The McMahon Correspondences (for the Arabs) and Balfour Declaration (for the Jews) were conflicting by many accounts, and definitely by the perceptions of the people in the area. As it were, the British technically also promised to have parts of Palestine be an international trusteeship, but reneged on that promise with the French (the Sykes-Picot Agreement). So the area was not uninhabited. The question of "stole" is a matter of opinion.

The Zionists definitely came to an area that was already populated, and populated it with Jews. The goal was definitely to establish a Jewish state, and some Jews recognized it would be at the expense of the national aspirations of the Palestinians in the area already. Many Jews also bought land, seeing it as a way of both establishing their claim and creating a connection to the area, though they only succeeded in buying 6% of the land by 1947 or so. Keep in mind, however, that private land ownership does not guarantee sovereignty. I do not "own" a nation even if I own 50% of the private land one can buy, nor does it give me more of a vote in pretty much any democracy. Part of the problem, however, was the system which the Arab peasants in Palestine had set up to farm. Because of Ottoman land law, which the Ottomans began to attempt to enforce more strictly over the later years of its existence, one had to have a title to the land they worked, which was intended to boost tax revenues. However, much land was worked by poor peasants who couldn't afford the taxes or the title, and worked communally to farm areas that might've been state-owned land, for example. What these peasants did was to ask richer Arab landlords to buy the land and assume the title, assuming they would be allowed to stay and work the land as they always had. This was also beneficial to the peasants because their names thereby avoided the tax rolls, meaning they were not drafted for the Ottoman military, something they were not eager to have to do.

Keeping this in mind, when the Zionists did manage to buy land (the Ottomans attempted to stop land sales to Jews at times, as did the British, neither was effective) from an Arab, difficult as it was with restrictions and also a bad stigma for the Arabs who sold to Jews, they mostly bought the land from absentee landowners who were richer and simply allowed peasants to work the land. However, the Jews who assumed the title wanted to work the land themselves, an integral part of the Labor Zionist cause (which was the major strand and which would become the political party dominating Israeli politics for the first 30 years of its existence). So when these Jews pushed the peasants off, wanting to use Jewish labor (though some did choose to use Arab labor instead, which resulted in cheaper labor) to establish a separate Jewish economy and prove the Jews could establish their own state, the Arabs were displaced. There was nothing illegal, in a sense, about this. It cannot be called "thievery", but it is unfortunate and it was based on an ideology of excluding Arabs from Jewish jobs, which was justified by Zionists as a move towards both establishing a Jewish state and as a way of avoiding conflict with Arabs who had already reacted very poorly to Jewish settlers from the get-go (accounts as early as 1885, from the first settlers in Palestine, not necessarily even Zionists, document Arab attacks due to tensions with Arab neighbors due to other factors in the Ottoman Empire). Jews were not exactly paragons of virtue either, and did attack and refuse to work with Arabs very early on too, as documented notably by Cultural Zionist Ahad Ha'am in 1891.

Was it stolen? That's a difficult question to answer, and it really depends who you ask. The Zionists were not there with ancestral roots, for the most part, that clung to the land in recent times, but Jews felt a spiritual and ancient connection to the Holy Land embodied by their religion, which called for a return to Eretz Yisrael (as it is called by Jews) to herald the coming of the messiah. Further, the partition plan of 1947, created by the UN (I'll get to that later) promised the Jews and Muslims and Christians that it would keep Jerusalem under international trusteeship, meaning no one group had a sovereign claim to the area and all had free reign to visit as they pleased. It split up the area so that 55% went to the Jews (though much of that was the southern Negev desert and considered unusable, but Jews planned to settle there anyways and "make the desert bloom"), and 45% to an Arab state. There were fewer Jews, but the reasoning behind a larger area was not only because the Jews were happier to settle the desert slowly (though the Arabs wanted the Negev too), but also because there was expected to be a huge influx of Jews following the establishment of Israel, including by Holocaust survivors. This indeed happened: in the 3-4 years after Israel was established (1949-1952, after the war ended) there were over 680,000 immigrants, which doubled the population of Israel and posed a very big problem for Israel as far as accommodating them, many of whom had left behind what wealth they had and had to get jobs and new lives/homes.

So, what was the population like in 1947, according to Partition Plan estimates? The plan purported to divide up Christians, Muslims, and Jews. In the Jewish state, there were to be roughly 500,000 Jews and 400,000 Arabs. In the Arab state, there were to be roughly 720,000 Arabs, and a token Jewish population of less than 25,000. Jerusalem, kept separate, would have roughly 100,000 Jews and Arabs apiece. So Jews would establish a state in areas they formed the majority in, would have room for the expected influx of immigrants, and Jerusalem would be international.