Does turning the air conditioning on and skipping its "fan" setting damage the air conditioner?
Typically the blower motor or fan on a forced air AC or HVAC system only run when the AC is running (this is also referred to as the "auto" mode). Turning the fan to the on position only runs the fan, but not necessarily the AC itself, giving you some air circulation, without the extra power required to run the AC compressor.

With a window unit, you also have the option to allow outside air in with the fan, which can cool a warm room if it's cooler outside, without running the AC at all. The fan is also useful for homes where some rooms get especially hot between the AC cycles, so you can constantly run the fan instead of lowering the temperature on the thermostat, but if you do this for extended periods, I'd suspect it actually increases power consumption rather than decreases it.

As for whether you would damage an AC by skipping the fan setting, that would be highly unlikely since the fan automatically turns on when the AC is running. What you can do to damage an AC is rapidly turn it on and back off, since this can result in a pressure differential in the coolant lines that could prevent the compressor from starting normally, burning out the motor, and it could be in a state where the lubricating oils in the coolant are not at the compressor motor where it's needed (this is also why you don't turn a refrigerator on immediately after it's been tilted from shipping).

As for why the fan setting is found between off and cool on some thermostats, this is likely for convenience. It's either off, you want fresh air, or you want fresh air and cooling.