But was Israel/the idea of it already existing around this time?

So, the idea of a "Jewish state" had been something that had been discussed for a long time in Europe. The most famous proponent of this idea was a man named Theodor Herzl, who ultimately came to believe that the Jews would never be safe in Europe after the Dreyfus Affair (an incident where a French-Jewish captain was convicted for allegedly spying for Germany; his case was hugely controversial at the time and a lot of people considered it a major miscarriage of justice/episode of clear antisemitic bias) and the election of the far-right, openly antisemitic mayor of Vienna, Karl Lueger. Zionism also gained currency in the global Jewish community throughout the latter half of the 19th century, with the First Zionist Congress meeting in 1897. Zionists disagreed on where this new Jewish state ought to be; some people favoured Palestine, but others thought about establishing a colony in Africa, for example.

Another important factor in why Palestine was the way it was in 1948 was the collapse of the Ottoman Empire after WWI. The Ottomans had controlled most of the Middle East for centuries, and after they collapsed, someone needed to step in to fill the vacuum. The winning European powers of WWI sat down and decided to carve up the Middle East into various spheres of influence, with Palestine coming under British control (this is called Mandatory Palestine). The British allowed some waves of Jewish immigration, though it's also true that many people came illegally. The reason why many Jews felt that Palestine had been promised to them is because of a document called the Balfour Declaration, where Britain promised to support (and here's the exact wording) "the establishment in Palestine of a national home for the Jewish people", though they also said that "nothing shall be done which may prejudice the civil and religious rights of non-Jewish communities in Palestine". The actual declaration is really short and you can read it [here] (https://en.wikipedia.org/wiki/Balfour_Declaration#/media/File:Balfour_declaration_unmarked.jpg).

As for what had been promised to Palestine, a lot of people see the Balfour Declaration as contradicting the promises that were made to Hussein bin Ali, the Sharif of Mecca, in a correspondence between him and a British official named McMahon. McMahon was trying to convince Hussein to lead an uprising against the Ottomans during WWI and promised him that "Great Britain is prepared to recognize and support the independence of the Arabs within the territories in the limits and boundaries proposed by the Sherif of Mecca". You can read the McMahon-Hussein correspondence in full [here] (https://www.jewishvirtuallibrary.org/the-hussein-mcmahon-correspondence-july-1915-august-1916), it's also not very long. There have been long debates about what specifically was promised and how those promises may have been interpreted, but the Arabs felt betrayed both by the Balfour Declaration and the Sykes-Picot Agreement, which divided much of the Middle East into British and French territory, and accused Britain of violating its agreement. Hussein was so upset he refused to ratify the Treaty of Versailles in 1919.

For the next two decades, Britain now had to rule an increasingly chaotic and violent Palestine, with regular clashes between the Jewish and Arab communities there. The number of Jews also continually expanded as many fled Europe, since most non-European countries (for example, the US and Canada) either refused to accept refugees or put very limiting quotas (Canada, for example, admitted fewer than 5000). With mounting violence throughout the 1920s and 1930s, Britain put the Mandate to a commission called the Peel Commission, which declared that the Mandate was unworkable and recommended partition. See [here] (https://upload.wikimedia.org/wikipedia/commons/a/aa/PeelMap.png) for their proposed map. Both sides had issues with the proposed settlement (it would limit Jewish immigration, which Jewish leaders opposed) and obviously, Palestinians were not thrilled about the idea of having their land partitioned. Some Palestinian leaders were willing to consider it, but it was ultimately rejected. Ten years later, the UN made its own plan for partition (this is the one you referred to in the original question), which was also rejected by the Palestinian side. The Palestinian side ultimately rejected the UN proposal because it allocated the Jews of Palestine more than half the land when they were only about a third of the population. At this point, the British were determined to end the Mandate regardless and left in 1948, after which Jewish communities immediately declared a state of Israel, kicking off the 1948 Arab-Israeli war. Most Arab countries were united against Israel, but Israel ultimately won the war, forcing many Palestinians to flee their homes (this was called the Naqba, and is one of the major waves of Palestinian displacement).

Another major wave of Palestinian displacement is the 1967 war, which you mentioned (also called the six-day war). The fighting continued because Israel had majorly expanded its territory during the war, taking the Sinai from Egypt, the Golan Heights from Syria, and the West Bank from Jordan. Israel started settling this new territory, which obviously angered all of the countries that it had taken it from, and both sides engaged in regular cross-border raids. Israel and Egypt fought a war of attrition over the Sinai for the next few years, which was majorly ramped up when Nasser (Egyptian president) died and was replaced with Sadat, who really prioritized getting the Sinai back. Also, in Syria, Hafez al-Assad ceased power, who was also determined to get back the Golan Heights, setting the stage for the 1973 Arab-Israeli war. Syria and Egypt wanted their own territory back, which is why they kept fighting with Israel. Egypt ultimately regained the Sinai in the 1974 peace agreement, after which Sadat did end up making peace with Israel.

As for your question about whether the region could have been better divided, it's hard to say. The UN proposal definitely could've better reflected the demographics of the region; it definitely proposed giving more land to the Jewish settlers than was proportional. On the other hand, there are some major sticking points that don't seem to have good solutions. A major one is the status of Jerusalem, which is holy to both Muslims and Jews. A lot of people on both sides maintain that the city is (or should be) indivisible, and that it shouldn't be split into two. If Jerusalem is indivisible, who should control it? Israel? Palestine? The UN? If you divide it, there's questions about which parts should belong to who. The other major sticking point (and there are many, many of them) is about the right of return for Palestinians who have been displaced throughout the conflict. Palestinian leaders say that it's absolutely non-negotiable, and the Israelis absolutely refuse to accept it.

I hope I answered some of your questions! Please let me know if anything's not clear, I know I jumped around a lot trying to get to all of them.