What kind of dog does not bite or bark?

The only kinds of dogs that don't are the stuffed toy dogs.

There are dogs called "barkless", for example the Basenji, but they really do bark, just differently. Barking can be controlled with positive reinforcement training like other obedience skills, and by letting them know by your actions and the focus of your attention, that you are on the job and protecting the house (or car or child or whatever) whenever there is a loud strange sound or other bark-provoking situation.

They will only raise a ruckus if it looks like no one else is minding the store. If you let them think that, you will be turning over the leadership of the pack to them, you will no longer be the alpha dog in charge, and that is a very slippery slope. Keeping yourself looking strong and clearly in charge by your body language and reactions to situations, will also make them relax and behave, knowing you've got it covered.

As for biting, any dog will bite if they are not trained not to, it is a part of puppy play that they need. The play teaches them how hard is too hard during the socialization process (best in first ten weeks of life) with their litter mates and other pack members. If they do not learn this as puppies, then re-training later is more difficult, although a necessary part of any obedience training.

Any time a puppy even touches teeth to human skin, the human should squeal just like another dog would if bitten by the puppy. Watch a mother dog when her puppies bite her ear or pull her tail, she overreacts and they learn that is not something to do, or they learn from her response with her teeth how hard is right for "Gotcha!" play. You should squeal like it really hurts, even if you barely feel it, if teeth touch skin.

Having dogs who behave like you want them to is a result of training, which all companion dogs and dog owners should have. Except in dogs bred specifically for the guarding traits, these behaviors are learned, they are not breed specific.