a) During evaporation of a liquid the molecules of liquid having high kinetic energy move out off the surface of the liquid leaving behind the molecules with low kinetic energy. In this way the temperature of the liquid decrease and is becomes less as compare to the environment, so to compensate this loss, heat from the environment is absorbed by the liquid and temperature of the surrounding decreases that causes the cooling effect.
Evaporation results in cooling because the particles of liquid which undergoes evaporation draws the latent heat of vaporisation from other particles which is placed in contact to it and thus introduces cooling effect.
b) If humidity is high, then air is already saturated with water vapor's, i.e., it has a lot of water vapor's. Therefore, it will not take more water vapour easily. Hence, rate of evaporation decreases.
c) Sponge has minute holes in which air is trapped. The material is also not so rigid. On pressing this, air is expelled out, that is why it can be compressed but it is solid as it has a definite shape and volume and it does not change its shape unless compressed.