How does cold air move through a room

As mentioned in the comments, air from a fan feels colder because of two reasons. Firstly, the air is cooler than your body so as it passes, your body heat transfers in part to the air, which is then carried away. Second, as the air passes, it evaporates moisture on your skin, which takes absorbs heat in the process.

As for the propagation of cold temperatures in general. If you put an ice block out on the counter, the air around it will get colder and this coldness will seem to radiate outward. This is not because the air molecules are passing on a lack of vibration, it is because they are passing their heat to the colder molecules of the ice block. The excited, warm air around the ice block bumps into the molecules of the ice block and transfers energy to it. This causes that air to lose heat and be colder. Then the air immediately around that transfers heat to the first layer of air and loses heat itself. Repeat ad nauseam. This is all because objects tend to want to be in thermal equilibrium. To make everything the same temperature, heat tends to flow towards colder areas, which can seem like cold flowing outwards to warmer areas.

In short, what seems like cold flowing outward is actually heat flowing inward.

A fan turned on in a closed hot room will eventually increase the temperature of the room because the energy from the fan will turn into kinetic energy of the molecules plus direct heat from the inherent inefficiency of the motor. A 300 watt fan is like a 300 heater eventually, so the kinetic energy of the molecules will increase at the end of the story.

The cooling felt is due to the air flow established by the impetus of the blades. The air flow passing over human skin increases heat transport from skin contact (if temperature less than 36C or so) and also evaporation of perspiration. Over 37C only extra evaporation provides cooling.

The molecules within the air flow have the collective motion of the mass in the flow, which does not change the microscopic kinetic energy which defines the temperature of the moving mass. As I said above eventually all the energy dissipated will turn to extra heat in a closed room.

The temperature of the air didn't change. What you are experiencing is an increase in rate of evaporation from your skin - and this evaporation is what makes you feel cooler.

Why does the rate of evaporation increase? Well - evaporation is a balance between water molecules going from liquid to vapor, and from vapor to liquid. In a stationary air environment, water vapor concentration close to the skin will be greater - slowing down the rate of evaporation. When you blow "fresh" air (not so moist) against your skin, you can start cooling down more quickly.

There is also a very minor effect that the Bernoulli effect means that air pressure for moving air is slightly lower - I think that is secondary though.