Yes, you have to purchase it separately.
The size is as follows: Product Depth (in.) 0.75 Product Height (in.) 1.56 Product Length (in.) 4 Product Width (in.) 4
That would be too far. I'd put them about 5 feet apart maximum unless your rod is extremely rigid.
Yes. It is much easier to install when disassembled and is very easy .to reassemble
Each hook comes in two pieces and its distance from the wall can be adjusted somewhat. It _looks_ like you'll be able to get 5 inches clearance, but I'm not positive.