Can a dental infection cause vertigo?

Yes. I have seen vertigo show up from a dental infection before, but very rarely and typically only in severe cases.
Yes, rarely. Due to the complexity of the oral/maxillofacial region, infections can travel to various sites of the head and neck. An upper molar infection, or even a sinus infection could potentially cause enough swelling to affect the ear and cause symptoms of vertigo.
Yes. The ear controls your sense of balance as well as hearing. The inner ear contains a small fluid-filled structure called the labyrinth, which helps the brain find your center of balance. Damage or an infection to this area can cause dizziness or vertigo. In some cases, an infection in the teeth can spread to the nerves of the ears, leading to lightheadedness. Read more: http://www.Livestrong.Com/a.
Possible but rare. I've seen vertigo in a few patients from dental infection but it's rare.