In terms of wire size, 18 AWG is sufficient electrically (the router thingy uses roughly 1.5 amps per their web page), but I'd go with 14 gauge or larger simply due to the wire being more physically robust. Be sure it's fused appropriately where it connects into the 12V system.
The antenna you list doesn't need a ground plane per its documentation. The maximum thickness of the panel (roof, in this case) it can mount on is only 15mm, so mounting it to a box seems a good plan. I doubt there would be any noticeable difference in performance one way or the other.
Frankly, the whole setup seems to me to be overkill for individual or family use, but I suppose that's not really any of my business.
