My dog ate some papaya flesh along with a few seeds — are the seeds dangerous?

The ripe flesh is safe, but you should try to prevent him from eating the seeds in the future. Papaya seeds contain trace amounts of cyanogenic compounds, which can be problematic in very large quantities. A few seeds accidentally swallowed by a medium or large dog are unlikely to cause harm, but it's a good practice to always remove them. If your dog is very small or eats a large handful of seeds, watch for any signs of stomach upset and contact your own vet if you're concerned.