Is it feasible to cycle at South Island during June-August? Will it snow? How cold can it be? What's special gears for the bike? How worse the weather can be?
It does snow, but not everywhere, so there are still plenty of options for cyclists who don't mind the cold. Temperatures get pretty low at night so you might be less inclined to want to say in a tent, but there are lots of nice, clear, sunny days so in other respects this is a good time for it.