Evaporation of liquid produces cooling because liquid takes away the heat from the surroundings, thereby producing a cooling effect.
If humidity is high, then air is already saturated with water vapour, i.e. it has a lot of water vapour. Therefore, it will not take more water vapour easily. Hence, rate of evaporation decreases. (2)
A 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 does not change its shape unless compressed.