They took Walt for the same reason they take children in general, which is the same reason Mother took the two babies in the last episode - to mold them into being a potential "protector" of the island. That's the whole reason the Others are there at all - Jacob brings them there because he is looking for someone to take over as the island's protector. I think that is exceedingly clear at this point.
The pregnancy question doesn't specifically have anything to do with the Others, so I'm not sure why you're lumping that in there.
They live on an island inhabited by a smoke monster because Jacob brings them there ... duh.
Dharma Initiative - they are a group founded by Hanso, whose distant relative was on the Black Rock and wrote about the island in a journal that was somehow found and passed down through the Hanso family. The exact mechanics of this sequence of events won't be explained, but I don't think it's important enough to warrant it. Needless to say, Hanso found out about the special qualities of the island and funded a research group to go exploit it. He probably created the Lamp Post station to find the island - how he figured out how to do that, I'm not sure, and I don't care. But as far as what the DI actually IS, that's explained. The island has enough seemingly magical, psychological things happening on it (seeing dead people etc.) that running behavioral experiments seems perfectly natural.
Why was everyone brought to the island? Jacob brought them there in his search for a replacement. IT'S BEEN ANSWERED.
EDIT: And the cabin - pretty sure that, although the cabin was originally Jacob's, by Season 3 the cabin was actually inhabited by smoke monster/man in black/Jacob's brother. Ilana mentioned in season 5 that Jacob hadn't been there in a while, and that it was being used by "someone else". That's when she torched it. The "someone else" is certainly Man in Black, given the fact that Christian Shepherd occupied it and convinced Claire to stay on the island, and as we all know, she turned out to be on MIB's side by the end. So the "help me" was probably Man in Black talking to Locke, which is very interesting given the fact that Locke DOES turn out to help Man in Black by dying and giving MIB the opportunity to assume Locke's form.