Greetings,
The fan motor is an electronically commutated (brushless) type. Any delay between application of power and beginning of rotation is most likely caused by a soft-start mechanism intended to ensure that the power supply has reached a stable state. While a 6 second delay may be somewhat excessive for these purposes, it is not necessarily abnormal. Such a power-on delay is likely built in and not easily changed, but I would expect a greater degree of responsiveness to changes in control inputs.
Small DC fans of this type are fairly consistent in their interface and control requirements, a discussion of which can be found here. The operating manual for the fan is here. It consists mostly of boilerplate warnings, but helpfully indicates which wire is which and indicates that the tachometer output is in fact of the typical open-collector type.
Note that the typical pattern calls for a digital PWM control input, rather than a continuous 0~10V control signal. While this control input will likely consume a current on the order of a few mA or less, the resulting power consumption is usually insignificant compared to the amount of power required to run the motor itself.