My dog and my cat usually get along pretty well, but my dog knows who's boss. About half the time I go into the garage (their home), I see my dog sharing the pillow with my cat. The other half of the time, my dog's gotten booted.
As far as hunting goes, my dog couldn't catch a flea. He chases after birds and other dogs all the time, and used to chase cars. Neither one wishes to hurt the object in question, but my cat makes a much better natural hunter. She catches mice a lot and just plays with them until they die. She looks disappointed when they actually die though, as they're not as amusing just lying there on the ground. I went out into the garage one time, and my cat was chasing around this mouse for awhile. She'd just pick it up in her mouth and throw it around or paw at it or something. After about five minutes of watching that, it got a little old, so when the mouse ran by me I kicked it across the garage into the door, and that rather hindered it's mobility, so the cat got disappointed. She still played with it until it died and then some though. I think it's because we don't give her anything else to do.