Will my hoarseness improve if I restrict intake of cold drinks and cold foods?

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. There is no relationship with hoarseness and food, drink or cold intake. It occurs when the vocal cord are inflammed and/or not working properly. Infections, allergies, or overuse ( too much screaming, yelling or talking) are common causes. Treatment depends upon the exact nature of the cause.
No. Hoarseness is due to swelling of the vocal cords. Rest and avoidance of irrritants like cigarette smoke will help heal your problem.