Question 1
Why do we divide by the number of data points (N)? I think it's done to minimize the error being back-propagated, but can't we just don't do that and instead decrease the learning rate to decrease the "learning" instead?
Question 2
For classification tasks, would it work to apply a softmax activation function to the last (output) layer and then calculate the error using the Mean Squared Error formula? (For the sake of simplicity, cross-entropy is harder compared to MSE)