Possible. There are several factors that are related to dry mouth and using a dehumidifier can make a dry mouth worse. Mouth breathing can increase dry mouth. Many medications will cause dry mouth. Alcohol, even in mouth rinses can also make dry mouth worse. A dry mouth can lead to tooth decay so it is important to talk to your dentist and doctors about this and develop a strategy to manage this.
Could be a factor. Especially if you are a mouth breather. If you have nasal congestion or obstruction of the nasal passages, you are more likely to breathe through your mouth, which can dry out the oral mucosa. Also, you lose the beneficial humidification of air that the nasal passages provide. The dehumidifier is probably not the cause, but could be a factor. See your doctor to check your nasal passages.
Very possible. There can be other factors, especially your breathing. Do you mouth breathe, meaning are your nasal passages blocking causing you to breathe more through your mouth. Allergies, etc. Do you sleep and wake up with a very dry mouth. May want to consult dr.
Dehumidifier. Anything that takes humidity out of the air in a room can cause a dryness in the room leading to dry mouth, dry sinuses etc. Better to use a humidifier if you are worried about this.