Training the Discriminative Model in Generative Adversarial Neural Network
In normal GANs, there are no labels, the training is completely unsupervised.

The role of the discriminator is to tell apart samples generated by the generator from those taken from the training dataset. The training dataset is just a bunch of images. The discriminator is trained to output 0 for data generated by the generator (i.e. fake data) and 1 for real data (so the discriminator has a single output). This should answer points 1 and 2.

The training of discriminator and generator takes place alternatively in a loop: first we train the discriminator, then the generator, then the discriminator again, etc. It is possible (and common) to train the discriminator a few times per each time we train the generator. This should answer point 3.

It is also possible to use labels, but not in the way you were suggesting. When labels are used, we have Conditional GANs (https://arxiv.org/abs/1411.1784). In this case, the label is supplied as input to both the generator and the discriminator. The generator has to generate data that is associated to the supplied label. The discriminator has to tell apart fake data from real data, given the label.