A question, if I play seasick on the Kings of the North Caitlen Stark, being played during the player action box after attackers are declared, does she remain in my opponent's hand or does she go to the discard pile? Also, if she remains in his hand can simply replay her during this same player action?

This is essentially the same question that was asked about triggering/canceling Banner for the Storm from hand in this thread.

So if I understand it correctly, there is no purpose in trying to cancel an effect such as this because the opponent can just attempt to play the same card over and over until you run out of cancels. Now does this still work if one of the revealed plots is Fear of Winter. Or does the first cancelled attempt consume the one card played from from hand?

It is pointless to cancel Catelyn's ability, even if Fear of Winter is revealed. This is because Catelyn technically only leaves her owner's hand if the triggered effect goes through. If her triggered effect is cancelled, then it's as if she never left the hand, meaning a card was never played from hand.

Ron said:

A canceled effect never actually happened. So in the case of a canceled Catelyn vs. Fear of Winter, the Stark player never actually put a card into play. The GJ player, on the other hand, HAS played Seasick.

Note that a canceled event when Fear of Winter is out is different because with events, playing the card from hand is part of the cost - therefore, even with the effect canceled it counts against Fear of Winter. Catelyn puts the character into play as part of the effect - which never happens if canceled.

Oh, and ultimately, yes: canceling the new Catelyn is an exercise in futility.