Is there a risk of inhaling foreign body accidentally through the nose into the lungs?

Yes. The nose is connected to the lungs by the trachea, so there is a risk of aspirating a foreign body from the nose into the lungs. Signs of foreign body aspiration include sudden choking, breathing difficulty, coughing and wheezing. If you suspect your child of aspirating a foreign body, call your doctor right away. This could be a life-threatening condition.
Yes. This is probably the worst possible complication of a nasal foreign body and thus the most compelling reason to remove a nasal foriegn body.
Yes. It is possible to inhale a foreign object through the nose and into the lungs, but it is not easy to do. The path up the nose, through the back of the nose, down the throat, and into the trachea (a tube between the throat and the lungs) is a crooked path that has some narrow areas. Luckily, most foreign bodies get stopped along the way towards the lungs.
Yes. Yes. If a foreign body is in the nose, depending on its size, it may get into the lungs. If your child has a foreign body in their nose, they should be seen by a physician immediately.