Statement (A) : Water can be boiled in a thin paper box without even charring the paper 

Statement (B) : In winter, woolen clothes keep us warmer compared to cotton clothes.

Statement A is true. It is because heat supplied to the paper does not appreciably heat up the paper. Paper conducts away most of the heat, that it receives, to water. Secondly to burn a substance, the temperature of the substance should be made greater than its ignition temperature. Here conducting property of the paper does not let its temperature to rise up to ignition temperature. Before that time, water boils.

Statement B is true. Woollen clothes have pockets where air gets trapped in them. Trapped air takes heat from our body. But since the trapped air cannot move away, its heat is not removed from it (remember heat is a bad conductor). Only way to remove heat from air is to let it go. But air is not let to go.