Palmach Museum was created in 2000.

Palmach's motto is 'לפקודה תמיד אנחנו'.

Yes

It is unclear what is meant by "Palestinian Involvement". The Jewish Militias of Palestine, like the Haganah and Palmach (who at that time were called Palestinian Jews) actively contributed to the British War Effort against the Nazis. The Palestinian Arabs did not assist the British War effort, although they were involved in supporting the Grand Mufti of Jerusalem who was a close friend of the Nazis and Fascists in Europe.

The Modern Nation of Israel came into existence through the philosophical aspirations of Zionists, the gradual settlement of Halutzim (Pioneers) in the British Mandate of Palestine, the Militias of the Haganah, Palmach, and Irgun, the UN Partition Plan of 1947 and its vote to sanction a modern Jewish State, the declaration of Statehood by Ben Gurion in 1948, and the War of Independence (1948-1949). All of these things went into the Establishment of the Modern Nation of Israel.

If you meant to ask about Israelites as opposed to Israelis, see the Related Question below.

As concerns Israeli women (i.e. women settled in the Yishuv that would become the Modern State of Israel), women were equals in that 19th and 20th century enterprise. Although they were less involved politically than key male Zionists, Israeli women helped to build many of the early Kibbutzim and served in the Palmach and the Haganah Jewish militias. As most of the early Yishuv was secular, women's roles and treatments were indistinguishable to those afforded men.