“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.
Imagine your puppy running out the front door. You call him, waving a treat. But he'd rather chase a squirrel into the road than come back to munch on a treat. In addition to the obvious danger of Puppy getting hit by a car, he learns that he doesn't have to listen to you. He learns that he's in charge of what he decides to do and what he decides not to do.
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.
Your young puppy is totally reliant and dependent on you to help him habituate and fit into our human, domesticated world. Your guidance and leadership will determine what path his life takes and what type of dog he will become. During puppyhood you play the lead role and are responsible for shaping the character, temperament and behavior habits that your dog will carry throughout his life. Your puppy's future is in your capable hands...
You can teach your puppy at home and I'll help you. My puppy training book is called Respect Training for Puppies: 30 Seconds to a Calm, Polite, Well-Behaved Puppy. I'll show you my proven step-by-step training schedule for teaching your puppy all the words he needs to know, plus consistent household rules and routines, housebreaking, crate training, acceptance of being handled, calmness, gentleness, and general obedience training.