Why a double flute?
The short answer: because that is what 80% of the router bits in the world are, and that is what you're going to find readily available. (not trying to be a smart ass here.)
The long answer: You could use a single flute, but it will be more difficult to find "off the shelf" and the cost difference vs. a DF (double flute) will be marginal. You won't easily find a flush trim bit with more than 2 flutes. 3F bits are generally only found on solid carbide tooling, custom carbide-tipped tooling, or highly specialized carbide-tipped router bits. 4 plus flutes are generally used in the metalworking industry on solid carbide and high speed steel endmills. For this post, we will only be referring to woodworking tools and applications as metalworking is whole different animal. Contrary to popular belief, more flutes does NOT equal better edge finish; that is determined by other characteristics of the tool, mostly tool geometry. Certain bits *may* provide a better finish with more flutes, but it is not a rule of thumb. Adding flutes requires you to increase your feed (inches per minute that the bit travels through the work) and speed (rpm of spindle). The feeds and speeds required for more than 2 flutes is too fast to be safely done with a hand router. 3+ flute router bits are generally for CNC routers only. You'll also get more life out of a DF than a SF because you have double the cutting edges to wear before the tool needs sharpening. Never throw your router bits away when dull, they can be sharpened for a fraction of the cost of a new bit. The company I work for charges 4 bucks to sharpen a flush trim bit.
Hope I didn't go too tool nerd on you, these are the kinds of questions I answer on a daily basis and I can get carried away

Yes, you can use a flush trim bit for your trackball hole, but you'll need to bore a hole slightly larger than the diameter of your flush trim bit. Move the bit in a counterclockwise direction as you route the circle. Don't force it, but don't move too slowly either or you might get chip re-melt
As for which one to pick of the 8 or so available, that will depend on your router. If you're using a laminate router you'll need 1/4" Diameter (OD) x 1" Cutting Edge Length (CEL) x 1/4" Shank. If you're using a full size, two-handed router, you'll want a 1/2" OD x 1" CEL x 1/2" Shk. Once you determine that, eliminate all the bits with up or down shear. Now chose a bottom bearing bit and not a top bearing bit (that's a template/patternmaker bit, but I see they show up when you search "flush trim" on the Rockler site.) That should leave you with the bit you need.
Why the 1" CEL? The general rule of thumb is to use a CEL slightly longer than the thickness of your material. It will be difficult to find a flush trim bit with a CEL shorter than 1". Additionally, buying a 1" will cover most of your needs as a hobbyist, and as you wear one portion of your cutting edge, you can adjust your bit and use a "new" portion of your carbide, improving the life or your bit.