You are absolutelly right, but I do not understand the difference between your claim and mine. Maybe I am missing something

Do not worry, there is nothing you can do that would lead to that! Just check that your results make sense

Oops! You are absolutelly right! My confidence interval are of course for the 0 class!!

Please remember that neural networks are very complex structures with lots of parametres to tune, therefore you should not expect it to "learn" without huge numbers of examples. This is the reason why simpler models are often preferred when samples are not huge.