OP asked some specific questions, but I don't see how this answer addresses any of them.

Thanks for your explanation! But I don't fully understand how this is related to calibration and uncertainty estimation in neural networks. What I mean is: in those fields we want the neural network to not just have the prediciton for the data, but a distribution over the parameters and outputs. This way, the NN may output "The image is a cat with probability p=0.8". But I don't see how the NLL measures this

I updated the questions with two examples

What if p(x | Y = C_k) or p(Y = C_k) are zero? Then we have a_k = log(0)

Thank you very much. And what if $f$ depends on $\phi$ ?