How does an image classifier select relevant features?
There is no single answer.  A cascade is a very simple idea: it basically represents a bunch of classifiers, applied sequentially.

You are free to decide how each individual cascade will work.  You could design the cascade so that every classifier uses the same set of features.  Or, you could design the cascade so that each classifier has a different subset of features.  They're both considered cascades.

In the standard cascades I've seen used in computer vision, there is one set of features.  Each stage in the cascade is a single classifier which is trained on the entire feature vector.  Part of the training involves selecting a subset of the features (perhaps only a few features) to use at that stage of the cascade.  The training process is responsible for that, and there are many different algorithms for training out there.  One example is ADAboost, with decision stumps as the weak classifier; you could take a look at that to see how it works.