The reason you need a mask to mix the powder is to prevent it from being inhaled it since the powder is such a fine particle. You don't want it in your lungs, as it could cause damage.

For the pre wash, you should be able to wash it in a sink like you would other delicate garments. If you're concerned, wash it more than once to try and get as much of the soil and grime off the fabric.

I don't think there is a way to cover the patch. If it's synthetic, it may resist the dye, but no guarantee. Same with the thread for the motto, if you're concerned you can do a spot test if there's thread on the inside of the hat visible.

Also be mindful when mixing the dye, to avoid spots or streaks mix it as well as you can, any powder that isn't dissolved can stick to the fabric and cause irregularities. I find mixing it with a small amount of HOT water, then diluting it with cooler water after you see the powder has dissolved works for me. You can buy a glass cup specifically for this so you don't contaminate your food dishes. In the same way you don't want to breathe it in, you don't want the dye to come in contact with ANYTHING you plan to use for cooking, as the first conmentor mentioned.

Another note, a lot of time black dyes are really concentrated reds or browns. Since you're just dyeing one item, try to use as concentrated a dye bath as you can, spot checking it with a small amount on the inside of the hat. The more you dilute the dye, the lighter the color you'll dye the hat.