Every dog needs to learn to walk on a leash. Besides the fact that most areas have leash laws, there will be times when keeping your dog on a leash is for his own safety. Learn how to introduce your dog or puppy to the leash, then teach him how to walk properly on the leash, even beside you on a bike. A loose leash walk teaches your dog not to pull or lunge when on the leash, making the experience more enjoyable for both you and your dog.
“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.
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.
It typically takes 4-6 months for a puppy to be fully house trained, but some puppies may take up to a year. Size can be a predictor. For instance, smaller breeds have smaller bladders and higher metabolisms and require more frequent trips outside. Your puppy's previous living conditions are another predictor. You may find that you need to help your puppy break old habits in order to establish more desirable ones.