I had joined a monthly card swap hosted by Amy Phelps last year that was amazing! The themes were pretty open and varied and it was fun having so many cards on had. Kind of in keeping with this thread though, I was getting quite a few "praying for you" cards which I just don't use. I've now opted to going to the swaps threads on SCS and choosing which swaps I want to participate in. This way I can avoid ones where the theme or sentiment isn't quite me and I can choose how many swaps I'll do and what kind of time committment I'll have. I would really suggest checking out the swap board here.
As for how to incorporate cats, what about using cool design stamps to create a "wallpaper" or "upholstry" backgound for the cat? I realize you can't do that for every single card but it could be cool. Do you have a cat that could look out a "window"? Then you could use landscape stamps to create the view out of the window. I only have one cat stamp and he's up on his back legs, batting at something. I like using him with butterfly punches.
As my best friend says, "Stamping....it's cheaper than therapy" to which I add, "Not by much".
My mom was the proud recipient of my first handmade card. Cost $200.