“Down” can be taught very similarly to “sit.” You can wait for your dog to lie down (beginning in a boring, small room such as a bathroom can help) and capture the behavior by reinforcing your dog with a treat when he lies down, giving him his release cue to stand back up (and encouragement with a lure if needed) and then waiting for him to lie down again. When he is quickly lying down after standing up, you can begin saying “down” right before he does so.
If he's an older dog, he's probably used to his name; however, changing it isn't out of the question. If he's from a shelter, they may neglect to tell you that he has a temporary name assigned to him by staff. If he's from a breeder, he'll come to you with a long name, which you may want to shorten, or change. And if he's coming out of an abusive situation, a new name may represent a fresh start. But we're lucky: dogs are extremely adaptable. And soon enough, if you use it consistently, he will respond to his new name.
Help him relax when he comes home. When your puppy gets home, give him a warm hot water bottle and put a ticking clock near his sleeping area. This imitates the heat and heartbeat of his litter mates and will soothe him in his new environment. This may be even more important for a new dog from a busy, loud shelter who's had a rough time early on. Whatever you can do to help him get comfortable in his new home will be good for both of you.
Once your dog can stay in a sit for several seconds, you can begin adding distance. Place him in a sit and say “stay,” take one step back, then step back to the pup, give a treat, and your release word. Continue building in steps, keeping it easy enough that your dog can stay successful. Practice both facing him and walking away with your back turned (which is more realistic).
Next, drop a treat on the floor near you. As soon as your puppy finishes the treat on the ground, say his name again. When he looks up, give him another treat. Repeat this a couple of times until you can begin tossing the treat a little further away, and he can turn around to face you when you say his name. Avoid repeating your puppy’s name; saying it too often when he doesn’t respond makes it easier for him to ignore it. Instead, move closer to your puppy and go back to a step where he can be successful at responding to his name the first time.