I'll try to be brief, but it's a challenge.

1a. Do the Palestinians have any legitimate claim to an independent state?

I'll answer this "no," but for somewhat technical reasons. "Legitimate" seems too related to law or legal authority, and I don't believe there's any legal authority through which the Palestinians have a "legitimate claim" to an independent state. Someone else could answer it "yes" by pointing to U.N. general assembly resolutions, but I don't accept the U.N. as any kind of legal authority. It's reasonable for Palestinians to want to have an independent state, and maybe that's more in the spirit of what you're asking. I'm skeptical that this is the primary desire of Palestinians.

1b. Should the state of Palestine never exist?

If a state can be established that isn't regularly attacking Israel, I don't have an objection. I have serious doubts about whether such a state can live in peace with Israel. I don't think much would change if it were recognized as a state. There would still be regular attacks, responses, and recriminations.

2. Do Jews have the ultimate right (as in it can pre-empt all other claims) to Jerusalem? (I understand the city to be claimed by Palestinians and Israelis as a capital.) That is, does either group have a sole claim to the city, or should/can it exist as a capital of two independent nations?

I don't think Jews have an ultimate right to Jerusalem. I can't think of any situation where I would say an ethnic or religious group has a "right" to some land. As a practical matter, Israel will not give it up. (It's as unrealistic as Turkey giving up Istanbul.) However, the deal offered in 2000 shows some Israelis are willing to give up some parts of Jerusalem to be a capital of Palestine. I'm skeptical that this would work in practice, but I have no objection to it being tried.

3) Do you believe the expansion of Jewish settlements into what the Palestinians claim is Palestinian lands is justified? (Or do you believe Palestinians have no legitimate land claims, so the question is moot?)

While I don't believe the Palestinians have "legitimate" land claims, I can still try to answer the first question. The expansion of Jewish settlements into new parts of the West Bank makes the effective land area for a future Palestinian state smaller. I think this is a reasonable price the Palestinians should pay for decades of committing, encouraging and rewarding terrorism. Imagine there were a magical force field that determined the border, and that every time there were a Palestinian terrorist attack on Israel that force field expanded outward by one meter. I would consider that a good thing. I would feel differently if Palestinians discouraged terrorism and punished terrorists.

4. And finally, if there should be a two-state solution, what border should Israel occupy now? Was the 1947 plan that was devised by the UN inherently flawed or biased against Israel, or was it only the Arab wars against Israel that made the original plan unworkable?

The peace deal offered in 2000 gives a realistic idea: something close to the 1967 borders but with land swaps to account for Jerusalem and the "facts on the ground." I don't know if the 1947 partition plan was workable in 1947, but I seriously don't think it is today. There has been too much population growth and movement since then, in addition to other issues.