Why and how did Jack get sick (appendicitis)?
This was really bugging Rose, the way it happened at a key time, and that it happened on this island where people get better instead of sick. Thing is, it had to happen. In one timeline Jack had appendicitis as an 8-year-old. And we know that certain major life events have to have occurred in any timeline a person enters.

What is 'the war,' and why does Widmore say it is coming? Has it not already been going on?
Well, we're smack in the middle of it now, with Smokey going all freight train on the Temple. We still can't be 100% certain of which side is "the wrong side," but are pretty sure that means Smokey.

Does Kate ever get to fulfill her purpose for going back to the island - to find Claire?
Yep.

What is Richard Alpert?
Richard is basically Jacob's "prophet" to the Others, the one who serves as a go-between for Jacob and the people he has recruited that fit his belief in humanity's basic goodness. He is a Spanish man from the mid-1800s who arrived as a slave on the Black Rock and was saved, oddly enough, by Smokey, who then tried to get him to kill Jacob by promising that Richard could be with his dead wife Isabella again (hey, did anyone ever stop to think that MIB was going to make good on this promise to Richard and Sayid by killing them, sending them off to the afterlife to see Isabella and Nadia?). Richard doesn't age because of Jacob's touch/gift, which was granted so that Richard could avoid paying for his sins in death.

How did Christian's body become reanimated?
Better put, this question should now read, where did Christian's body go? Because we pretty much know MIB does not "possess" bodies, he merely takes the form of them, as he has done with Locke. We're pretty certain, though, that whenever we saw an animated Christian Shephard that it was Smokey. There are problems with this, though: 1) like we said, where is his body (we know where Locke's is; where's Christian's? 2) Ghost Christian has appeared off-island, which is seemingly something Smokey can't do (go off-island, as he complains about being trapped there an unable to leave). But Christian appeared to Jack in the hospital one night after the O6 returned (and the "smoke" detector malfunctioned!), and he appeared on the freighter telling Michael his work was done. Was Jacob perhaps appearing as Christian in these scenes? Has Jacob ever appeared as Christian?

PARTIALLY ANSWERED

What is 'the Monster'?
What we know: The MIB/Smoke Monster are the same entity. He lacks a body, but can appear as those who are dead. He has described himself as a man, and talked about having had an insane mother. He seems to like killing people. He can not penetrate ash circles, or perhaps he can once there is no longer a Temple Master.
What we don't know: His name, from where he came, why he wants so badly to leave, where he hopes to go, why he requires followers, what happened to his body, what his job as a "security system for the temple" was all about (or whether that was a lie or a mistaken claim), or why Ben/The Others were able to summon it.

Who are the Adam & Eve skeletons, and what is the significance of the black and white stones lying with them?
What we know: One is male, one is female. They don't appear to have died of any trauma. We know Rose and Bernard flashed back to the 70s and intended to live our their lives there. The stones could be clues that this was our favorite interracial couple. There are anagrams that suggest this is the Nadlers (their last name). During this season's visit to the caves, Hurley postulates that these could be some of their 815 compadres.
What we don't know: Anything for sure. Could be anyone. But it sure feels like Rose and Bernard. And anything else at this point is going to seem kinda strange. Also, I don't think we'll be going back to the caves. Why else have Jack & Hurley stop there on their way to the Lighthouse if not to kinda-sorta bring this loose end to something of a conclusion.