You absolutely can use pet hair from a cat or a dog for flies.
I have a pair of carding brushes that I use to mix dubbing. Works better than any coffee grinder or air system especially for long fibers. They are used to pull apart and align wool from sheep and llamas so the fibers can be spun into wool yarn on a spinning wheel.
They are great for small amounts of dubbing and you can titrate the amounts of different fur to get just the mixture you want.
Dog brushes can be used but carding brushes work much better and faster.