On this card it reads "Kneel one of your House Stark characters to choose and kill a character witg equal or lower strength." Does this mean your own character or your opponents? Plus, does that character that you kneel stay that way until all the way until the next round during the stand phase?

Unless it says "Kill a House Stark character" or "Kill a character you control" it means ANY character.

Once knelt, characters stay knelt until the beginning of the Standing Phase, or until a card effect makes them stand.