It's best to keep them separated. By nature, however lovely they are, dogs are predators. It's normal behaviour for them to chase, hunt and kill little animals/birds. It may be play now but it will become real. As for the hen, her playing dead may not be 'play'. It may be a survival instinct many prey animals have. Freeze and hope the predator leaves you alone. It's not malicious of your dog, it's just natural behaviour. Some dogs are bred to use soft mouths with birds so they don't damage them. Most gun dogs can hold a bird gently -labs, retrievers, spaniels etc. They were bred to flush and/or retrieve birds for hunters and not damage or eat them. That instinct can't be untrained.

What kind of dog is he?