It would depend on the amounts of water and sugar
Yes, sugar can dissolve in a clear glass of cold water without stirring, but it will dissolve more slowly compared to if the water were warm. The process can be accelerated by stirring or heating the water.
You can increase the temperature of the water.
by heating the water slowly and by stirring using a spoon, you can dissolve salt and sugar.
What temperature is the tap water? A little will dissolve in room temperature tap water, but if you raise the temperature of the water more sugar will dissolve. If you boil the sugar, it will break down into two simpler sugars and won't precipitate when the water is cooled.
This depends on: - mass of sugar - granulation of sugar - volume of water - stirring (and intensity of stirring, stirrer type) - type and geometry of the beaker - exact temperature of water
Vinegar is usually a solution 9 % of acetic acid in water. The solubility difference of sugar between water and vinegar is not significant at equal temperature, granulation of sugar and volume, without stirring.
Stirring makes it faster for the sugar to completely dissolve in the tea.
That depends on the relative quantities of sugar and water, the water temperature (more definitely than merely "cold"), and any agitation (stirring).
These effects are the same when dissolving anything into anything. Stirring, or motion, is the key. Stirring plays a role because it is motion in the solvent at the macro level. The temperature of the solvent (water) plays a role because the higher the temperature the more motion in the volume at the molecular level.
Factors that cause sugar to dissolve faster include increasing the surface area of the sugar (finer crystals dissolve faster), stirring or agitating the solution, raising the temperature of the solvent (hot water dissolves sugar faster than cold water), and increasing the concentration of the solvent (higher concentration can dissolve more sugar).
The sugar cubes soften when placed in the water and begin to dissolve. The sugar cube in the hot water dissolves the fastest. Sugar will dissolve faster when you stir the solution quickly because the act of stirring increases kinetic energy which increases the temperature.
Sugar can typically dissolve in one cup of water as long as the water is warm or hot. Stirring the water also helps to fully dissolve the sugar more quickly.