What characteristics must a solvent being used for recrystallization of vanillin should have to obtain the best results? One characteristics I could think of is it has to be cold.

Answer 1

The solid must have high solubility at high temperature and much lower solubility at low temperature in the solvent or mixture of solvents chosen.
The solid is dissolved at high temperature in the least volume of solvent, then cooled off. Unless the solution becomes supersaturated, the "excess" solute is crystallized and can be separated.
Vanillin can be dissolved in boiling water and re-crystalized by cooling in an ice bath.