The advantages and disadvantages are closely linked. The obvious advantage is not having to make expensive traditional masks. But the downside is that nanosphere lithography will only let you make those shapes that your nanospheres will allow (usually hexagonal arrays). If you are considering this technique, I assume that's acceptable for you.
Monodispersity of the spheres is important too. Though monodisperse nanospheres are fairly easy to make or buy, there is always some variation between spheres, which is going to influence the packing of the spheres and will lead to small deviations from the theoretical perfect ordering. And if you need some very specific size of particle which is not available as a standard product, that will take more time and money to make (getting a very precise size can involve some trial and error).
Finally, there is the issue of dimensional stability. Wet nanospheres usually are slightly bigger than dry ones, due to attached solvent, stabilising groups, and internally bound solvent. Temperature changes may also lead to them expanding or shrinking. So, especially when drying them using fairly aggressive methods (strong heating and/or vacuum), drying cracks in the layer become likely.
1) only two kind of arrays can be produced: randomly distributed spheres for low concentration, or hexagonal close packing if the concentration is high enough to form a monolayer of spheres.
2) the size of the final nanostructures will depend on the diameter of the initial spheres, and only a limited number of diameters are comercially available. You,can try etching the initial spheres to reduce their diameter, but this may lead to additional problems: hardening of the spheres, damaging the substrate or the template, etc.
The main advantage, of course, is that you can pattern large areas at low cost.