Kate and Christian - what did I miss?

From what I recall, the last time we see Kate is when Sawyer and her
and the baby are sleeping on their way back to the Losties at the
beach.

Then Sawyer wakes up, she's gone and he finds the baby.

That's the last we saw of the 3 of them, in that episode, and haven't
seen them since, correct?

So now all of a sudden Kate is in the cabin. OK, so she's dead.

So.. I don't get it. I know we don't know the answers, but we all are
assuming Christian is or isn't Jacob?

And Kate is there because..... she's Christian's daughter? or to help/
confuse Locke?

And of course Locke doesn't know who Christian is anyway..... unless
it was explained to him.

Signed,
a little confused