Why is it impossible to build an air filter that lowers the average velocity of the air molecules in a house? 
Imagine a special air ﬁlter placed in a window of a house. The tiny holes in the ﬁlter allow only air molecules moving faster than a certain speed to exit the house, and allow only air molecules moving slower than that speed to enter the house from outside. Explain why such an air ﬁlter would cool the house, and why the second law of thermodynamics makes building such a ﬁlter an impossible task.

If the house is an isolated system(no heat exchange between the house and the environment), then by the second law of thermodynamics, the entropy of the house cannot decrease. But in this process, the average velocity of the molecules decreaces, thus the entropy decreases.
My questions are
1) Is there any heat exchange between the house and the environment?
2) Is it true that when the average velocity of the molecules decreases, the entropy of the house decreases?
 A: Actually that's not an impossible task as long as you don't constrain the problem by not allowing energy input to the filter. The problem is the famous Maxwell 's Demon, but in the end you have to pay the demon. His efforts don't come free. The Hilsch tube, originally thought to house the demon fails the challenge as it takes excessive energy to separate hot and cold molecules. It only appears to work. The second law says no free lunch.
A: 1) Is there any heat exchange between the house and the environment?
If the molecules within the house decrease their average speed, there is most certainly an exchange of energy between the house and the outside environment.  The Law of Conservation of Energy provides that energy neither can be created nor destroyed.  The filter you described succeeds in transferring energy from inside to outside the house.
2) Is it true that when the average velocity of the molecules decreases, the entropy of the house decreases?
The Second Law of Thermodynamics says that within an isolated system, entropy can only increase.  As the house in your problem is NOT an isolated system, it makes no sense to look at entropy only inside the house.  One must look at entropy of the entire system, which includes the outside.  If the Second Law is true, it would be impossible to operate such a filter without expenditure of energy, which must come from the entire system, increasing the entropy of the entire system.
