Is it right that if it warms a few degrees to evaporate, it roughly cools the air around it be a few degrees? Curious how much air is cooled how much how quickly.
The short answer to this is yes, but it's a lot more than a few. The long answer is that the question as stated misses an important point. When 1 kg of liquid air warms one degree it cools 1 kg of gaseous air 1 degree, that much is true, the heat is conserved. However, when a substance boils (or condenses) it absorbs (or releases) a large amount of heat. This quantity is known as Heat (Enthalpy) of Vaporization.
For air, I'll model it as having a heat of vaporization intermediate both oxygen and nitrogen, specifically 80% N2 + 20% O2. This is a pretty rough calculation so it shouldn't matter too much. A quick google search says 6.82 kJ/mol for O2 and 5.6 kg/mol for N2, so our value for air is 5.844 kJ/mol. Fine, how much does a mol of air weigh? I'll continue with the 80%/20% model and I get 28.81 g/mol. To boil 1 kg of air we need 202.85 kJ of energy.
The last piece of the puzzle is relating that energy to a change in temperature of air, this is done by using the specific heat capacity. Expressed as kJ/Kkg it is the amount of heat it takes to raise the temp of 1 kg of a substance 1 kelvin (Celsius). Air has a specific heat capacity of 1.005 kJ/Kkg (at constant pressure, which is the case here and I don't want to make this any longer) so boiling 1 kg of air will lower the temp of 201.84 kg of air 1 degree C or 1 kg of air 201.84 degrees C
As an aside this also tells us that we'd get 1.005 kJ of energy for changing the temp of the liquid air 1 degree, so you can see that there's a huge energy cost for boiling compared to changing the temp, in this case ~200x vs. raising the temp 1 degree.
The last thing I want to note is that a liquid kg of air occupies approximately 1.15 L, whereas (at ambient conditions) 1 kg of gaseous air occupies 820 L of volume. A 713x fold increase. Boiling just 1 kg of liquid air can affect huge volumes of space.
Note that this is all thermodynamics and doesn't really speak to how fast things are happening. However, the energies involved are large enough that this is going to be pretty fucking fast. I'm not even going to try to calculate it since there are just too many unknowns, but a lake (or even a pond) of liquid air is nothing to sneeze at.