We investigate the structural and dynamical properties of the transcriptional regulatory network of the Yeast Saccharomyces cerevisiae and compare it with two "unbiased" ensembles: one obtained by reshuffling the edges and the other generated by mimicking the transcriptional regulation mechanism within the cell. Both ensembles reproduce the degree distributions (the first-by construction-exactly and the second approximately), degree-degree correlations and the k-core structure observed in Yeast. An exceptionally large dynamically relevant core network found in Yeast in comparison with the second ensemble points to a strong bias towards a collective organization which is achieved by subtle modifications in the network's degree distributions. We use a Boolean model of regulatory dynamics with various classes of update functions to represent in vivo regulatory interactions. We find that the Yeast's core network has a qualitatively different behavior, accommodating on average multiple attractors unlike typical members of both reference ensembles which converge to a single dominant attractor. Finally, we investigate the robustness of the networks and find that the stability depends strongly on the used function class. The robustness measure is squeezed into a narrower band around the order-chaos boundary when Boolean inputs are required to be nonredundant on each node. However, the difference between the reference models and the Yeast's core is marginal, suggesting that the dynamically stable network elements are located mostly on the peripherals of the regulatory network. Consistently, the statistically significant three-node motifs in the dynamical core of Yeast turn out to be different from and less stable than those found in the full transcriptional regulatory network.