Effect of carbonated water on the rising of the dough?
This is rather simple. The carbon dioxide created by yeast does not generate new air bubbles in the dough - the yeast fills and enlarges existing air bubbles, that are in the dough due to the kneading, whipping of eggs etc.. If you add carbonated water to the dough, you create more air bubbles that the yeast can enlarge.

This should have no dramatic effect on the rising time; temperature and yeast amount will matter much more. But it strongly influences the final texture of the baked good, as more initial air bubbles before the rising will result in a finer and tenderer product.