Why does the air we blow/exhale out from our mouths change from hot to cold depending on the size of the opening we make with our mouth? Why does the air we blow/exhale out from our mouths change from hot to cold depending on the size of the opening we make with our mouth?
It's not just a subtle difference, but significant in my opinion. I'm inclined to discredit the notion that it's just a matter of speed because I can blow fast with an open mouth and still, it's hot; and blow slow with an almost closed (tighter) mouth and again, it's cold.
 A: It is mainly due to air entrainment.
If you blow through a tight mouth, there is smaller volume of air but a higher velocity. This pulls in and mixes with a lot of ambient air (Venturi or Bernoulli effect) - in fact typically the air stream is only 40% body warmth and 60% ambient so it will be markedly colder. As an experiment you pucker and blow through a tube held to your mouth, this excludes the ambient air and you will get reduced airflow but at the higher temperature again.
With a wide mouth there is hardly any air entrainment. Its temperature will be almost same as its temperature in your lungs which is higher than the ambient temperature.
A: I don't think it's quite that physical. The pressures involved aren't that high
When you breathe slowly on your hand the air is war, moist and 37C so feels warm compared to the surroundings, if you blow through a small opening the flow of air increases the cooling and evaporation from your skin.
A: I think most reasons given above are false. It is our common experience that cold air comes when we exhale through a narrow opening of mouth. (Experience it yourself). It is due to adiabatic expansion of air. When a gas is allowed to expand suddenly, it does so by absorbing heat energy. When air is suddenly exhaled out into a larger volume through the narrow opening, air undergoes adiabatic expansion. When we place our hand near the out flowing air heat energy is being absorbed from our hand. Hence we feel cold. Opposite is in the case when a gas is compressed. Heat energy is liberated. (try this by tightly closing your mouth with your hand and exhaling air out into a small volume compressing it. You feel that the air is hot). So the exact reason is the adiabatic expansion or compression according to the size of the mouth.I wonder if the case is different in any other cold countries. In India it is as explained above. 
A: The total sense of hot or cold is set by the integration of the response of all the sensors in the skin, +- evenly distributed, involved in the experiment.    
The sense of hot is proportional to the area and the sense of cold is  proportional to the perimeter*$\delta r$ , for the same pressure (see the answer of user1355).
The outcome of the experiment can be described by the ratio hot/cold  or _(area)/(perimeter*$\delta r$)_ $=\pi R^2/2 \delta r \pi R=R/2\delta r$, i.e proportional to $R$ .
Let Experiment open mouth have radius $10 R$ and Experiment almost closed mouth have radius $R$.
The ratio of the outcomes of the two experiments (open/closed) is 10 .
If someone has doubts, and I do have, that the perimeter can contribute significantly to a colder sensation than the overall outcome becames more contrasted (ratio of areas, proportional to $R^2$) 
and the  above example will give the value 100.
An array of thermometers can set the question, imo. One measure can be defined as the sum of all the temperatures.  
The description in the question is correct, supported by experimentation (I did it ;), and have a physical and rational interpretation.  
