What exactly does generator produce in DCGANs?
Does it generate the set of the same image classes in the same order on each iteration? 


No, a basic generator in a GAN is typically fed a small random vector as input; e.g. a column of 100 gaussian samples, with mean 0, standard deviation 1. It then must use this random "embedding" in the feed forward network to produce output that is in the target distribution of real data that the discriminator is trying to assess it against.

So a GAN typically learns a mapping from an arbitrary relatively low dimensional space onto a higher-dimensional target space. DCGANs do this with image data.

The class of an image can be used as conditioning data, if it is fed as additional input to the the generator and discriminator. The sequence of image classes can be anything you like, and the GAN is not required to work in a particular sequence (unless perhaps you are working with a RNN-based GAN and are trying to learn to produce sequences).


  What exactly does generator produce in DCGANs?


It produces an output image depending on the input. The output is drawn from a distribution based on the training data, that the discriminator would struggle to distinguish from real data. If fed a random input vector, generated in same way as training stages, then the idealised output should appear to have been drawn equally randomly from the imagined population of all possible training data.