Thanks, all you cat lovers!
I used Wunder Under for all the pieces because that's what I had in the house. I didn't use any sort of stabilizer on the back of the block. I was in such a rush, it totally slipped my mind, but it worked ok anyway, and yes, it's just zigzag stitch.
I first outlined the eyes and nose with a fabric pen (I cut the eyes and nose out of my tracing paper pattern with an exacto knife, then placed it over the applique to I could get them in the right place). I knew I wouldn't want to stitch the black outline by hand. Then I hand embroidered the green eye fill, black eye slit, pink nose fill, nose bridge outline and mouth outlines. I didn't try to cover up the nose outline, because the real cat's nose actually has a dark outline.
This project almost didn't come off because I have so little grey fabric in my stash! The piece I settled on was in a bundle my mother won as a door prize at a quilt show. I don't like the print by itself, but it works here!
Thanks again for all the nice comments!