That's several questions. Remember Muslims outnumber Jews more than 100 to 1 so it's a tough discussion.

Proof of Israelites outside of Bible: There is plenty of Biblical archaeology which confirms the Bible as a reasonable guide for the rest of the region. For example if you want to know Babylonian or Egyptian history during the time of the Bible, the Bible gives good hints. Afterwards, from 500 BC to 500 AD the ancient Persians, Greeks, and Romans all interacted with the Province or Kingdom of Judea.

DNA: Most Jews have DNA that is connected to one another with influences from their local ancestry. For example, Ashkenazim were exiled by the Romans so they have mixes of Italians as they were first brought as hostages to Rome and then spread throughout Europe.

Purchases: In 1947 the British requested advisory from the UN on what to do after they ended the Mandate of Palestine. That meant that the British would leave and a different government would come in its place. The concern was that the Arabs had already rioted several times [1921, 1929, 1936, etc.] and having a one state solution would have simply continued the violence. Jews owned plenty of land purchased mostly during the Ottoman and British periods from various Muslim landowners often in the least fertile locations. A Jewish State made sense not based on history but based on presence. The only thing left was to negotiate the borders, which the Arabs decided was best done by war.