Lots of questions, and I'll do my best to be succinct.

I'm curious why the Jewish population was given modern day Isreal after the second world War.

If you take nothing else away from this, I think the most important thing you can gain from my answer is that the Jewish population was not given anything at all. It's true that some world powers supported them, and others opposed them, having a state in what is modern-day Israel. However, it was not given to them, and each inch of what comprises Israel today was paid for in tears, in sweat, in hard work, and eventually in blood.

From what I understand, that region was taken over by the Ottoman Empire, which was then conquered by the British, but was still effectively Muslim in culture.

This is true, in terms of majority population. There has always been a Jewish presence in the area, but it has been small outside of Jerusalem, which has fluctuated between Christians, Muslims, and Jews. The Ottoman Empire had controlled the area for hundreds of years, and before that Arab Empires, though there are minor interruptions and discrepancies due to rebellions and invasions throughout. Nevertheless, it was the Ottomans who held the area until 1917 or so, when the British captured the area (which they later gave determined borders, under the Mandate for Palestine).

This was not when Jews began to desire a state in the area. In fact, Jews had already begun immigrating to the area decades prior, many of them viewing it as a return to a home that they could finally enter, in order to one day set up a state or an autonomous zone. This was a period of self-determination and national identity, and Jews were no exception; they viewed the growth of these movements for autonomy for communities and groups as every bit as pertinent for them as for anyone else, if not moreso given the long history of the Jewish people and their perseverance in the face of persecution. In fact, the founder of political Zionism, the stream of thought that argued Jews deserved a state in modern-day Israel and self-determination as a people in that form, was active long before the British took control of the area from the Ottomans. Theodor Herzl, a Jew born in Vienna, wrote one of the most famous pamphlets (quite short) summarizing the Jewish position and laying out a basic plan for what Jews would do to establish that they could govern the area, operating under the belief that Jews would show they could govern and gradually would reach statehood themselves.

Herzl sought the approval of major powers, as well as their assistance in convincing the Ottomans to grant Jews a semi-autonomous province, or statehood, in the area of modern-day Israel. Despite garnering the support of some leaders, like those in Russia, Herzl died with his dream unrealized in the early 1900s, though not before he had built an organization that would be crucial in the push for Jewish statehood.

The Ottomans had been resistant to Jewish desires to build a state generally, but it was WWI that provided Jews with a new opening to push for autonomy and independence. During WWI, the British (for a variety of complex reasons, dealing with both the course of the war, antisemitism, philosemitism, and other strategic considerations) decided to issue the Balfour Declaration (this was 1917), which accepted and "viewed with favor" the idea of a Jewish "national home" in modern-day Israel. How big this would be was unclear; as the British helpfully noted around 1922 in what is often called the Churchill White Paper, the goal was not a "wholly Jewish Palestine" (by this point, the British defined Palestine as modern-day Israel, Gaza, and the West Bank, though it had been a more fluid region prior to this definition), but simply a Jewish national home "founded in" Palestine, which meant somewhere inside. The subtext was also that a national home did not necessarily mean a full state.

Nevertheless, the British were largely supportive, at least at first, of Jewish efforts towards statehood. They eventually became more reticent, but the Jewish cause had garnered the support of other major powers at the moments when the British backed down, eventually getting a nod of approval from the USSR and United States.

Throughout the entire period pre-Holocaust, Jews were pushing for a state. Before the Holocaust, there were already a large number of Jews in the area; immigration and natural growth from births had greatly increased the proportion of Jews in the area, to over 20% of the population.

It's important to note also that Jews were cognizant of this fact. By 1947, Jews were around 33% of the population, and did not ask for the entirety of the area. Indeed, Jewish leaders recognized the futility of that. Some of them called for attempting to control the entire area, but the vast majority of Jews believed this was pointless, and that the goal should be a viable state, and maybe someday Jews might be in control of the entire area, but this was unclear and impossible to know. Ben-Gurion, Israel's most prominent founding father and first Prime Minister, put it thusly in a meeting of the Jewish Agency Executive (then one of the top bodies of Jewish governance in the area) in 1938:

Ben-Gurion: The starting point for a solution of the question of the Arabs in the Jewish State is, in his view, the need to prepare the ground for an Arab—Jewish agreement; he supports [the establishment of] the Jewish State [on a small part of Palestine], not because he is satisfied with part of the country, but on the basis of the assumption that after we constitute a large force following the establishment of the state — we will cancel the partition [of the country between Jews and Arabs] and we will expand throughout the Land of Israel.

Question: By force as well?

Ben-Gurion: Through mutual understanding and Jewish-Arab agreement.

This, of course, may have changed after the Holocaust itself. Ben-Gurion's statement that he knew a Jewish state was not viable throughout the land at first and could only arrive by effectively making the Jewish position stronger with a state first in part of the land is in contrast to what he later spoke of. Chaim Weizmann, also an incredibly popular Jewish statesman at the time (also heavily responsible for getting the Balfour Declaration through and Israel's first President, though that's largely a ceremonial position) told the UN Special Committee on Palestine (UNSCOP, more on that later) that he favored a partition of the area into a Jewish and Arab state. UNSCOP then asked Ben-Gurion what he thought. He put it thusly:

I will tell you what we told the Government last year and this year while we believe and request that our right, at least to the Western part of Palestine should be granted in full and Western Palestine be made a Jewish State, we believe it is possible. We have a right to it, but we are willing to consider an offer of a Jewish State in an area which means less than the whole of Palestine. We will consider it.

In speeches to Jewish leaders, he was even more conciliatory, knowing likely that a public hardline position (we will only consider an offer of less than the whole and maybe won't take it) was never going to happen. To his wife, he called UNSCOP's recommendations "the beginning, indeed more than the beginning, of [our] salvation." In private conversations with UNSCOP before they recommended partition, he and other Zionist leaders said partition was their preference (clearly knowing full control was unlikely) and Ben-Gurion said that separation and partition into two states would mean "fertile cooperation between the two states."

CONTINUED IN PART TWO IN A RESPONSE TO MY OWN COMMENT

You're asking a lot of questions here! To start off, I'll refer you to this answer by u/ghostofherzl. (I would just generally read their answers- they're well written and clear and on a wide variety of topics about Israel.)