Question: I noticed when it snows and salt is melting the snow on the roads in the city, that the air temperature feels a lot cooler than the temperature indicated on the thermometer.
A 40 degree fahrenheit temperature feels like 30 degrees. Is it the moisture from melting snow or rain that makes the air temp feel ten degrees cooler or is it the salt brine (melted snow or ice and salt? Is it similar to adding salt to the ice in an Hand powered ice cream churn to make the cream container colder to make ice cream?

Answer: 
It depends on the exactness of your thermometer and your definition of ‘nearby’.

Since adding salt to ice causes the overall mixture to assume a liquid phase rather than the previous solid phase, melting enthalpy is required to liquefy the ice/salt mixture. This melting enthalpy is typically supplied by drawing heat from the surroundings, i.e. due to the melting everything gets colder. This, naturally, also draws heat from the surrounding (close) air. However, there are no *direct* effects on the surrounding air.

You can read more about the effect of salt added to ice in [this question](https://chemistry.stackexchange.com/q/5748).
No. It merely reduces the melting temperature of the ice. If you've ever noticed, salt water doesn't freeze until a much lower temperature than regular water. It is because of the same reason.