I will address your question on how the fan actually works and cools you. @sammy gerbil answered the other part about the noise. I can only add that you probably confuse the fan noise with the whistling in the ears when the wind is strong.
Fan creates a flow of air — i.e., increases the speed of the particles in particular direction. Naively, this would mean the increase in temperature.
But this is an ordered flow on a scale much larger and with speeds much lower than a typical particle speed (for scale: the speed of sound in air is about $300 m/s$ — more or less of the order of a typical molecule speed). So this flow is barely noticeable by individual molecule.
You can split the mass of air into pieces (small in terms of the flow but large in terms of air molecules), that are roughly comoving. In a frame of reference where a particular piece is stationary, the air won't be different from air in a room without a fan.
It turns out, the fan does not influence the temperature of the air directly (except for heat losses mentioned by sammy gerbil and heating through dispersion of flow energy). Different effects are important.
1. Mixing the air
For starters, consider a human in the room. Human is a power source that constantly produces heat (although at a small rate), so the air around the human will be slightly warmer than in other places. If you force this warmed-up air to move somewhere else, the human will feel a bit cooler.
2. Moving the water vapor
Sweating is a natural mechanism of the thermoregulation. When we sweat, we form a slim waterskin with large surface. Although in general water molecules stay together, the most energetic ones are able to break the bonds and fly away — and the more surface you have, the more molecules escape. When the most energetic molecules are gone, the total energy of the water is smaller — hence it becomes cooler.
However, if the air is filled with water vapor (for example, the same escaped molecules), evaporation becomes inefficient because while some molecules get loose, others come back. This is the place where the fan helps us: again, it moves the moist air away allowing us to evaporate the sweat.
But why fast molecules do not mix themselves? It turns out that molecules never travel far in a straight trajectory. They often collide and change direction. For example, the mean free path in the air is about $68 nm$ according to Wikipedia. So it would take really long for a particular molecule to cross the room by itself.
Regarding your idea of a flow in many directions: I think you are basically going to put more but less powerful fans in the room facing away from each other. That is probably inefficient because of the heat losses and inability to create a steady flow around the room that will circulate the vapor and hot gas.