You didn't mention how cold it gets where you live. I know some folks who consider 50 degrees F to be cold. I wouldn't think twice about a dog being outside for 30 minutes in that weather. On the other hand, if it dips below freezing, I wouldn't expect the dog to go out for any longer than it takes to potty, and then I'd bundle them up in a warm coat for a small dog.
I do know folks who have a kennel inside their fenced yard. It works well for them. However, the kennel is mostly to confine the dog when they are not around (it does sleep inside and is inside with the family when they are home). For the most part, the dogs do not play in the kennel. I would not expect your pup to exercise himself or play inside the kennel when you are not with him. An outside kennel could be a good solution if say, you need to be gone for a long period of time and there is noone to let the dog outside. It provides extra security from escapes, the kennel is easier to keep free of potentially dangerous items (say, sticks, or potentially poisonous plants). It should include a dog house for shelter and a water bowl. If you do this, I'd also get a run with a top on it. This will keep out predators like hawks or coyotes. With your medical condition, it could be used as safe potty spot for the puppy at times when he needs a bit more time outside. For instance, the last puppies I was around often pottied twice after eating- once immediately after, and then again 15-20 minutes after that.