Q1: In thermodynamics, you need to distinguish between your system and your surroundings. In this example, your system is the equilibrium between ammonium nitrate, ammonium ions, and nitrate ions. The surroundings is the water.
So, the reaction causes the temperature of the surroundings (the water) to fall, meaning that the system takes heat away from the surroundings. If the system takes heat away from the surroundings, then heat is going into the system. This means that heat is going into your reaction (i.e. your system) and therefore, the reaction is endothermic (ΔH > 0).
Q2: Absolutely correct. Since you must heat a liquid in order for it to evaporate, evaporation is an endothermic process (ΔH > 0). Also, since a gas is more disordered than a liquid, evaporation causes an increase in entropy (ΔS > 0).
Q3. Correct as well. Good job.