How many dinosaur genes are in a chicken genome?
Be careful when reading media as they tend to exaggerate. That Abzhanov paper is interesting; it will be better when it's published and the figures released. He does a lot of evo-devo work surrounding beak development. What they did in this paper was to inhibit in chick embryos two signalling pathways (which are actually conserved amongst all animals, not just birds and reptiles) so that their expression domain more closely matched that of extant reptiles (which lack medial activation, as opposed birds). They found that doing so results in premaxillary and palatine bones which resemble ancestral archosaurs. While incredibly interesting, this is a far cry from creating living chickens with dinosaur snouts.

To somewhat answer your question, chickens do not contain all of the genes that dinosaurs once had. They will have both gained and lost information. There will be much conservation in the same way that there is conservation between all organisms. For example, the Wnt pathway studied in this paper is conserved across all animals, from chickens to humans to placazoans to, presumably, dinosaurs. It's impossible to say how much has changed since we do not have any whole dinosaur genomes, but it's also important to note that it's not only which genes an organism has but also when and where they are expressed and how they interact.