Is it permissible to eat the bird caught by a hunting dog by holding the prey between its jaws, and is it permissible to eat game hunted by a rifle if the bird falls dead without being ritually slaughtered?

The Fatwa (definition)'s Summary:

If a trained dog inserts its claw or fang into the game, the bitten area is considered impure and must be washed seven times, one of them with earth, according to the Shafi’i and Hanbali schools. However, the more preponderant opinion is that washing it is not obligatory.

Hunting with a rifle is permissible if its owner finds the game already dead. If he finds it alive with a stable life, but is unable to slaughter it before it dies, it is permissible. However, if he finds it alive with a stable life and is able to slaughter it but fails to do so before it dies, then it is not permissible to eat it.