Welcome to the forum, Cando!
For the hat, if you find a pattern you like that will get the right gauge with doubled Shadow, go for it! If not, you can improvise a hat pattern by measuring your gauge with the yarn. For example, let's say you have a hat pattern written for worsted, which starts with k2p2 rib. Make a swatch of k2p2 with the Shadow and some appropriate needles. Then measure how many stitches you'll need to have the same width as the worsted hat. Cast on that many stitches, and you're good to go.
For the hank vs. ball, ABSOLUTELY wind it into a ball, unless you're really into driving yourself nuts. Trust me. The moment you start pulling from a hank, it turns into The Thing from 20,000 Fathoms. Get a center-pull ball if you can. There are tutorials online - I'm pretty sure Amy has a video about it. (Bless you, Amy!)
For the gauge swatch, I would go ahead and do them, then unravel them and reuse the yarn!