For the Wooden Dove, consider bird--evergreen. (The evergreen modifier is specifically for "wooden" pieces.) Then the bird--wings can go to the Flying Swallow.
For the Flying Cat, use tressym. (That's what a tressym is -- a cat with owl wings.)
For the Roaring Dog, dog--wind isn't quite a bad choice, but maybe dog--pow could work.
Addendum: With the recent update to my set, a piece that's "flying" can also be indicated with Element: Air as a (usually) foreground modifier.
A few icon recommendations/suggestions:
Addendum: With the recent update to my set, a piece that's "flying" can also be indicated with Element: Air as a (usually) foreground modifier.