Note, as of April 20 these issues have been resolved. You can fight and train on Facebook with the new Breeding Dome dragons.
Beauty, Magic, Chaos, Happy, Dream, and Soul element dragons appear to not be able to breed in the Breeding Mountain, Ultra Breeding Tree, Breeding Sanctuary, and Deus Breeding Nest.
Problems with Fighting on Facebook
Also note that while they can fight in iOS and Android, they can’t in Facebook. You can select them for your team, but they will not be able to be brought into battle. Likely they will also cause a glitch if selected for your Combat World team, causing problems for anyone who gets matched up against you. So be considerate and don’t leave them on your team on iOS or Android even!
Problems with Training on Facebook
Dragons with the new elements that are sent to the Training Center won’t be able to select any attacks to train. They may get stuck there, as there is no timer. Do not try to train them on Facebook! Training works fine on iOS and Android.