Regression equation for ordinal data
I think that what is influenced by the type of your independent variables is not the structure of your neural network but the encoding of your data. If your independent variables are ordinal, then you can map them to integer numbers or other ordered sets (and maybe normalize them to make them between 0 and 1). 

If your target variable is a label, it is a classification problem. Therefore, the neurons in the last layer in your neural network should have an activation function optimized for classification problems, like sigmoid. 

You also need a specific loss function, like binary crossentropy. There was a case study, however, that showed that for ordinal classification (when your target labels are ordinal), mean squared error works well.

You can take any activation functions for all hidden layers in your network as long as they are non-linear. The choice of a particular activation function will influence the rate of convergence of your learning algorithm but I don't see how a particular activation function in hidden layers may be preferable for a certain type of input variables, like if they are ordinal or not.