Usually takes a few days for a dog to adjust to new surroundings, so I wouldn't worry about that part. as for taking him outside I would keep him on a leash. If you have one of those extensible leads that would work as the dog could have more freedom but still be under your control. The one thing that makes me wonder is all that stretching he's doing when you call and the fact that he's lethargic. Was wondering if he could have a bladder infection or something, dogs with a uti stretch lot sometimes because of the discomfort. Maybe you could bring a sample in to the vet's? Just a thought.