Can bone get infected if you have deep hand cut?

Open fracture. With a deep cut that exposed bone can get infected if the wound get infected, it is not a typical scenario. However if the cut also cuts the bone such as with a saw, or there is devitalized tissue or there is an open fracture risk goes up. If the cut is due to a dirty source such as a bite or a farm injury and there is bone injury then that must be washed out properly to avoid severe bone infect.
Yes. Usually only after uncontrolled soft tissue infection.
Yes. Bone can be infected with deep cuts.The wound should be cleaned with running water and soap and check signs of infections.