No. Actually you should do the opposite and drink more liquids to ensure good hydration. Some people feel cold is better and others like hot- so pick whichever makes your throat feel better. A tbsp of honey in warm water can also soothe a sore throat and help with hoarseness. Hoarseness is caused by infection or overuse of your vocal cords.
No. Hoarseness has no relation to the type of food or drink. Actually fluids are helpful in most cases of hoarseness. The usual causes of hoarseness are viral, allergic or improper use of one's voice causing a strain on the vocal cords and possibly singer's nodules.
No. Hoarseness is due to swelling of the vocal cords. Rest and avoidance of irrritants like cigarette smoke will help heal your problem.