As for the first part:
Nukes have one great advantage over any other scheme that has been proposed. Every other suggested system requires matching orbits with the target before you can attempt to move it. So long as your guidance system is good enough a nuke doesn't need this.
Your target is closing on the Earth at 10 km/sec, you have to fly out there, then cancel your outbound velocity and add 10 km/sec back towards Earth before you've done anything--and at that point even with the biggest booster your payload is awfully small. Your nuke, however, doesn't care if it's going 20 km/sec relative to the target. So long as the fuse explodes it at the right altitude it does it's job.
There's also the little detail that nukes pack an awful lot of energy per kg and the number of kg we can get to the target is limited.
As for the second part of this question:
The reason breaking it up is a very bad thing is the square-cube law. The energy expends in all three dimensions, thus it drops off at the third power of distance. We live on the surface, though, the area goes up only at the square of the distance.
Look at what happens if we split a rock into 8 equal parts. Energy = 1/8, distance = 1/2, area = 1/4. Each piece does 1/4 the damage of the whole rock but there are 8 pieces for a total of twice the damage. (More energy is expended on blowing things up, less on making a crater.)
Breaking the rock up is the worst thing you can do unless you can break it small enough the pieces expend their energy in the atmosphere.