Thermal energy will flow from an object high temperature to an object of low one. In this case, the thermal energy will flow from object B to object A.
It will dissolve faster as the surface area is more so more particles collide ---> faster dissolving
On the other hand ammonia is a very dangerous chemical which has a pungent smell and effect the eyes of the user. Thus it kept always in the fume exhaust hood for storing and dispensing function.
The pH of ammonia buffer contains ammonium hydroxide (NH₄OH) and a salt of ammonia with a strong acid like (HCl) which produces, ammonium chloride (NH₄Cl) mixture. The evaporation rate of ammonia is so high at room temperature thus on opening of the buffer solution the ammonia get evaporated very fast and the concentration of ammonia decreases which affect the pH of the buffer solution.
Thus the reason to put ammonia buffer in fume hood is explained.