It jumped to New Caprica.
No idea. I'm guessing somewhere remote with a climate remarkably like Hawaii.
Though it would be awesome (but the set redressing would be too expensive to maintain on a TV budget) to have them pop-up in the Arctic or something (hence the polar bear research).
One old fan theory (though I think it would be in bad taste) was that the island somehow caused the Indian Ocean tsunami on December 26, 2004, which is about what time they're at now. Certainly displacing the amount of water the Island did upon leaving (and even worse upon returning) would cause a massive wave, as we saw with the ripple effect spreading out from where the Island disappeared.
So, having the Island pop up in the Indian Ocean is a possibility, and would help explain how Eko's brother's short range plane reached the Island (as well as the Black Rock ship a century and a half ago).
Why was Charlotte on the island?
I think they're trying to say she was Ben's childhood friend Annie (who also had red hair and freckles), though it's kind of a stretch to say she's supposed to be roughly the same age as Ben.
How does the frozen donkey wheel operate?
Very well.
It harnesses exotic matter to jump the island in space-time. The rabbit was jumped one hundred milliseconds in space-time, meaning it would briefly disappear and move slightly. I think the device temporarily removes you from space-time and allows the Earth to rotate underneath you, then you re-phase into normal space-time and reappear in a different location. However, since the shape of the Earth is not uniform, that can have drastic effects on the Island if you jump a great distance. Large parts of the Island can flood, or it can gain in height.
The old scifi rule is if you don't see the body, then there's no guarantee of death (and sometimes even having the body doesn't matter).
As soon as I saw the explosion from a distance and them not being able to go back and check for survivors, I felt for sure that Jin was still alive.
Where did Sayid take Hurley?
The afterworld
A world of never ending happiness
You can always see the sun, day or night
No idea, though if he knows Ben's plan to take them all back to the Island, that would make sense.
Jack, Kate, Sun, Aaron, Hurley, and Sayid for sure, though Claire's specter didn't want Aaron to go back.
Locke's body.
Possibly Desmond and Lepidus, but that's doubtful, and I don't think Ben even knows about their situation yet (though we might see Desmond again when Ben inevitably tracks down Penny to kill her).
Possibly Walt.
What bad things happened happened after the O6 left?
No idea. Maybe Jack not being there tipped the balance in the battle between science and faith between him and Locke and Locke screwed something up while he was in charge of the Others.
Maybe the Island and/or Jacob was punishing the people of the Island for not preventing them from leaving.
What's Desmond going to be up to?
Hiding out from Charles Widmore, and eventually on the run from Benjamin Linus as well when he comes to kill Penny (though they may not know that they should be afraid of him yet - however they might suspect since they should know that Widmore killed Ben's daughter).
I strongly suspect that Penny and Desmond are working with Sun to take down Charles Widmore in the future. Mr. Paik might have had insider knowledge that could be used against Widmore by Sun when she took over controlling interest of the company.
Are the Others going to merge with the 815 survivors?
I think so. Most of the redshirts among the 815 survivors are dead or separated from the Island now, and they need fresh blood to fill out the ranks next season.
I loved the scene on the boat though. "Hey, here's some lifejackets, swim your sorry asses to shore while we hop on this helicopter, chumps." They even held an open seat for Jin instead of giving one of the redshirts standing right there a seat.
What is the smoke monster??
I think it's the disembodied consciouses or souls of the people who have died on the Island.
What is the secondary protocol?
The one that comes after the first protocol.
Moving the Island from the Orchid Station was the secondary protocol, because that's how Keamy knew where Ben was going. The logo on the secondary protocol briefing dossier was that of the Orchid.
Why didn't Kate's stepdad/dad come to visit her?
Maybe he did. Maybe they had a falling out because of her being on the run from the law.