@Lollybags: Biting – Your puppy is biting you because he views you as a friend, not his master. As horrible as that might sound, you need to establish yourself as “master” first, and “friend” second. Sure, puppies will get mouthy with their mothers, especially when they get super energized, but she’ll quickly stop the behavior by getting up and leaving. Even if the puppy follows, you ignore the puppy, or, she will take him by the scruff/shoulders, and gently pin him to the ground until he settles down. You can imitate this by making a soft ‘claw’ motion with your hand and holding him lightly, but firmly to the ground until the puppy stops wiggling and trying to mouth you.
Also, make sure you’re not ‘squeaking’ when he bites you. OW! or NO! in a high pitched voice can trigger a ‘prey’ instinct in a dog, especially with terriers! They’ll get more excited and bite more! I know it is really common for people to suggest this, but it is not recommended to make any prey like sounds with a prey driven dog. Yes, their littermates will do this to let them know that he hurt them, but you are NOT a littermate, you’re his leader. Leaders are NOT to be chewed on. Puppy needs to learn this.
Be careful with the redirect of chew toys, as you don’t want to make your puppy posessive of chew toys! Make sure your puppy knows that YOU control the toys, but don’t ever take the toy from his mouth. If he has the toy you want, slowly approach him, stop a little ways away, and wait for him to drop the toy. Then, slowly take the toy. Practice this early, and you won’t be tugging shoes out of your grown dog’s mouth.
Does he bite your FI like he bites you? If not, observe how your FI behaves around your puppy. When he gets up in the morning does he rush over to puppy’s crate and play with him, or does he go about his morning more routinely? To establish that YOU are Mrs. Incharge, before you cuddle and give affection to your puppy, let him outside for his potty break, let him run around and exercise his energy away, and THEN great him with love and affection. Be calm, yet assertive, in the morning when you first greet the puppy. It might sound cruel, but it is very matter of fact, and you will be able to establish your dominance over puppy. This will also come in handy once puppy becomes dog. Dogs LIKE leadership.
Digging – To redirect unwanted digging, give puppy a place TOO dig! Get a sand/dirt box, bury things with interesting scents in the box (like a ball with treats in it). Let him use his nose to find treats! Encourage and praise him for digging in the right spot, but don’t scold him for being a dog. Like you said, he’s a terrior, it’s part of his breed trait to be a digger, it is who he is! You don’t want to discourage him to be who is he, but you must control where he digs. When you go outside, leash him, leave a scented object burried in the sand/dirt, and let him sniff it out, guiding him to the location of the box. Let him sniff through the sand/dirt, and praise him for digging there. Continue this pattern of “guided” digging for a while, then slowly start letting him off the leash in the backyard, monitor him, see heads over to the box and starts to dig.
Surprisingly, this digging will also help with the biting/chewing too! He’ll have an appropriate way to release his energy, and will not feel the need to release his energy by chewing or biting! Maybe even get him a little swimming pool or go on 30 minute runs with him each day. These are all great ways to drain a puppy’s energy, and it sounds like you’ve got a high energy dog.
I do want to make a note of the “marking” you’re talking about. I assume he is lifting his leg to ‘mark’ things? This is a good sign you need to be ready to take him to the vet to get him neutered. You’ll be able to prevent him from ever getting those nasty hormones in his body that makes him full of pent up energy from not being able to breed.