Answer:  

Dr. Kimberly Strong, who is studying atmospheric science, kindly answered the question above:
 
 
The simplest explanation for the decrease in temperature with altitude is as follows. 
 
Both the density and pressure of the atmosphere decrease with altitude. This is because atmospheric pressure depends on the weight of the overlying atmosphere. It decreases logarithmically with altitude, following something called the hydrostatic equation. Because of this decrease in pressure, a rising air parcel expands and cools (because it does "work" P dV on the surrounding atmosphere). The average rate of decrease is -6.5 degrees (K or C) per km in the lower atmosphere.
 
In fact, above about 10 km, the temperature starts to increase again due to the presence of the ozone layer. This absorbs solar ultraviolet radiation which leads to heating.
 
 

Variation of temperature with altitude   Courtesy of Aerospaceweb.org

 
 
Although Mt. Everest is high, its height is neglible compared to the distance between the Sun and the Earth, so the difference in solar heating between the top of the mountain and the valley does not compensate for the cooling of the atmosphere described above.
 
Reference
Most introductory atmospheric physics and chemistry textbooks describe the temperature profile of the atmosphere. Some examples are:
Chemistry of Atmospheres R.P. Wayne, CUP
Atmosphere, Weather, and Climate, R.G. Barry and R.J. Chorley, Methuen
 
 
Acknowledgement
We would like to thank Dr. Kimberly Strong for her expert answer.