Others may be able to add, but this is a good thread to start in. Make sure to check out the answer by /u/gingerkid1234 in that thread as well!

I'll note a couple of quick one-offs:

Among early Zionists there wasn't a general consensus that the Jewish state would be within Palestine, specifically.

There was no unanimity, but there was certainly a consensus. While there were debates around other options, the debates never went anywhere serious, and often concluded with a reaffirmation that the land of Israel was the only place Jews should seek. The most serious debate revolved around the Uganda Plan, which proposed giving an area in modern-day Kenya for a Jewish state, as well as the plan for a state in El Arish (the Sinai).

The latter arose as an option because of the sense Herzl had that the Ottomans would not consent to a Jewish area in Israel. However, while the British approved of the idea, the Ottoman Sultan (who still had some "nominal control" over the area) and the Egyptian government both objected. It seemed wholly impossible, inhospitable, and useless as an idea. The Uganda proposal is what followed, with the Guas Ngishu plateau near Nairobi being offered for a Jewish area, though they'd remain under British sovereignty. This area also seemed inhospitable, but that was because the white settlers already there under British control were extremely hostile to the idea, and Jews were not exactly warm to the idea either.

Herzl, apparently, didn't want either. Instead, his acceptance seemed more about getting recognition of having a right to a state, and appeasing the Great Powers by being accepting of anything they might give. Herzl was opposed in this by Max Nordau, another ardent Zionist. Isaiah Friedman recounts in "Theodor Herzl: Political Activity and Achievements" (published in Israel Studies) that Herzl received a letter from Leopold Greenberg saying that the Uganda Plan was beneficial to accept because it constituted a recognition of the Jewish people "as a Nation", which would persist and could not be revoked even "if East Africa is afterwards refused by us".

Herzl, in response, wrote, "We must obtain from the British government recognition of us as a nation, and the Charter should include the following phrase: 'creation of a Colonization company of the Jewish people'."

Nordau, who was Herzl's friend, was still opposed to the Uganda Plan, arguing that Jews would see a choice between East Africa and moving to America or Europe, and "prefer to migrate to America or Europe instead". He wrote that "The Zionist Movement would lose its raison d'etre and die a natural death" in that event. As you can see, it was quite well-known that the movement itself always revolved around Israel.

When Herzl explained that accepting the East Africa proposal was no more than "politically a Rishon le-Zion" (literally, "First to Zion") and would lead to the realization of Zionism's true goal of a Jewish state in Israel, Nordau became a wholehearted supporter of the plan.

Things changed not long after, because the Russian Minister of the Interior wrote a letter to Herzl promising to favor a Jewish state in Israel, and promising also to intercede with the Sultan. Suddenly, Jews had achieved a Great Power's recognition of Jewish nationhood.

When the Sixth Zionist Congress was held in 1903, Herzl expected celebration. He had a letter in hand from the British that de facto recognized the Jewish people as a nation through the Uganda Plan, and held the Russian letter sent just a week and a half earlier in his other hand, promising to intercede on behalf of Jews for a state in Israel. He opened the Congress with a speech saying, "There is no change and there will be no change in our attitude toward the Land of our Forefathers". But Herzl never explained, then, why he was providing information on the Uganda Plan, preferring not to reveal the tactic publicly (and also being sick). A month later, the plan died anyways, because Chamberlain resigned as British Colonial Secretary, and no more than a Commission to study the Uganda Plan ever got put into place (which Herzl expected would show how inhospitable the area was).

So despite these fleeting mentions of other proposals, even the most ardent supporters were doing so primarily as a way to gain recognition as a Nation, but planned to reject the offer ultimately. So realistically, there was a consensus among Zionists, though some tiny group may have seriously supported the proposal, meaning there was no unanimity.

I just mention this to establish just how universal Zionism was about the focus on Israel.

But why wasn't their land used rather than Palestinian?

The land wasn't Palestinian, it was Ottoman until the British took it over. The British Mandate for Palestine was the proposed location for a Jewish state from at least 1917, well before WWII. It was the desired area for a Jewish state among Zionists since at least the 1880s, when Zionism began to gain steam with the first waves of Jewish movements into the area.

Palestine was among the Axis powers, but they played a rather insignificant role in the greater war, and certainly were less culpable than Austria or Italy (or, why not even Japan if we're considering any Axis power as required to pay land as a cost of war).

Palestine was certainly not among the Axis powers. While prominent Palestinians defected to the Axis side, the most notable being Hajj Amin Al Husseini (the former head of the Arab Higher Committee, and the Mufti of Jerusalem), the propaganda that people like Husseini put out for the Nazis gained little favor with most Palestinians (at least, in terms of actions). Palestine was a British Mandate, under control of the British since at least the 1920s with the blessing of the League of Nations, and was designated to be a Jewish state that entire time. It was certainly not part of the Axis powers, and the population didn't show any significant favorability to the Axis, the few defectors like Husseini aside. It's accurate only to say that Jews in the British Mandate of Palestine were more willing to fight on behalf of the Allies and volunteered in greater numbers, but that doesn't mean that non-Jews were supporters of the Axis.

Interesting tidbit, the British Army formed the Jewish Brigade, mostly composed of Palestinian Jews from the commander-levels on down. They had a few thousand troops, and following the end of the war, some continued to hunt down Nazis. Many ended up back in Israel, and during the Israeli-Arab War of 1948, ended up being a crucial part of the IDF, providing much-needed military expertise and training just three years after the end of WWII.