It looks like this new Catelyn could be very useful if you build the deck around her. Instead of having Robb who can stand characters, you pile the power on the Blackfish to ensure you don't need to kneel Tullys in the first place. 

If you then start using the Stark tricks to shoehorn her into military challenges too and the opponent has a serious threat to contend with.

This card is now INSANE with the Tully cards like Edmure and Riverrun (and fast Eddy). Edmure pinching renown off opp or shifting renown of your own over to Cat, and riverrun adding to that. It's very easy to get her above ten strength comfortably. And of course not forgetting her own triggered action. Slap a Syrio's training on her and she is fully loaded. Beast card.

I don't believe the deck-type that favors this is as strong as the leaner, meaner Stark Fealty, but yes, Fat Cat can be scary good when things are rolling. The seven-coster affords a completely different game over Core Cat. You can win games with the Tully package (including this card as the centerpiece) w/o ever winning a challenge. Feels odd, but it works. 

can you use greywinds killing action and kill your own characters to trigger cats power gaining?

will the power gaining of cat also trigger when a copy/duplicate of a unique card gets removed? or is the save preventing a death?

If you save a character using a dupe or any other method. Then a character didn't die. so there is nothing for Cat to react to.

And yes there is nothing about cat that cares about how the character is killed of sacrificed. kill most of your board with wildfire if you like