Multi-class classification configuration 1) What are the appropriate activation and loss functions for multi-class classification problem?

Is it so that:


Up to 2 classes $\rightarrow$ Binary classification $\rightarrow$ Activation: Sigmoid $\rightarrow$ Loss: binary_crossentropy
From 3 classes $\rightarrow$ Multi-class classification $\rightarrow$ Activation: Softmax $\rightarrow$ Loss: categorical_crossentropy


If so then...


'Dogs and cats' are binary classification but 'dogs, cats and horses' multi-class classification?  
If we have 2 classes then we can't output probability ranges for prediction results?   


2) Is it so that for multi-class classification, the last dense layer must have a number of nodes equal to the number of classes?  

In the following example there are two classes (dogs and cats) and last dense layer has one node (why not 2 nodes?). If we have 3 or 10 classes then last node should have 3 or 10 nodes?
In my experience the activations are irrelevant to whether the classes are binary or multi-class. Their effectiveness depends on the kind of data you have. 

Furthermore, there is one node in the layer because you are doing a binary classification. In other words, when the value of the last node is low or below a threshold the class is 0 and 1 if otherwise(value of last node is high or above a thresold).

And yes, if there are more classes you need as many nodes in the last Dense Layer. You can also you 2 nodes when you only have two classes in your data and predicting the one with the highest score/output.