The girl that shot the arrow in the first episode is going to be important but probably not until next season or the one after. The boy that was pushed out the window will be somewhat important too down the road, but probably not as important as the girl will eventually be.
The king is going to die because, well, they aren't doing a ton to build depth to his character. I'm guessing he'll be used to move the plot along and his death, while not entirely shocking will set off some chain reaction of events
Ned is the hero of this season. They are spending a lot of time on him and his family. That means he has to go through some epic struggle that he overcomes to reveal a greater morality/lesson. I would say that he dies defending the king, but they haven't made me care enough about the king to accept that. I think people will want Ned to be king, but the king has that weasely son that is somehow going to mess things up for the good guys. I think Ned's kids will be compelled into action in future seasons, so something has to happen to Ned to get them there. I can see the first season ending with him getting captured by the king's killers and maybe in season two a couple of his sons try to rescue him.... But I'm getting ahead of myself here
Oh and something is up with the queen and those eggs. It's rather obnoxious how she keeps looking at them while she's getting railed. I have no clue why I should care about it, but they are being very intentional with how much camera time they are giving the eggs. I have a feeling that I'm not going to care for whatever they do with that part of the story line.