Of course, they don't all kill things unless trained.
Not true, beagles will kill rodents even if they are not trained
Beagles can live outside. Being bred to live in a pack, they would have many bodies to "nest" with, so maybe your question is more, is my pet beagle happy living outside because his breed was bred to hunt and they were kenneled outside? They love to play outside, walk, run, jump around in the snow. and as with any hair bearing animal, if your beagle is outside a protion of the day or all day, his system will conform to the colder weather by bulking up, as in a thickening of his coat. If your dog is inside all the time, they will not be used to the cold, and will suffer from the cold winter weather just as you would. If you plan to have your beagle out in the cold each day, you should provide a dog house that has a coule old blankets he can "nest" in to stay warm. If you live in an area that has below zero weather much of the winter, you should provide an insulated heated dog house.... I consider my 2 beagles house dogs. but they spend most of the day during winter while I am at work outside in the safe fenced yard, but have many sources of protection including thier "Dog Palace" when winter comes to visit :-D