18 or 16 gauge wire will be plenty. If you didnt know already, the size of the wire goes down as the gauge goes up.
You are running it at .7 amps. 14 is rated for 15 amps... twenty times what you are drawing through the wire. IMO, 12 gauge wire would be too large to solder onto the LEDs, would be harder to work with, and would be way over kill. If it were me doing this build, I would probably use 14 gauge since I have that laying around.
The fuse is just a means of circuit protection. If you decide to put one in, I would think that a 1 amp fuse would be adequate.