Is carbonated sugar-free soda bad for a children's teeth?

Yes. Remember acid is what ultimately causes the cavities. Some articles infer the diet sodas may be worse since they are more acidic. Acid is also the byproduct of sugar breakdown by the bacteria. That is also why you need to be careful not to switch to juices that are also acidic and rich in sugar.
Maybe. While the sugar-free soda doesn't have the main cause of cavities, the ph of the soda is very acidic - and that acid attacks teeth. Also, kids are getting used to drinking fizzy sweet drinks. Water or non-acidic drinks are way better habits to start. Root beer is the least acidic soda.
Yes. Even if sugar-free, the carbonation actually forms acid which eats away at the enamel of teeth.