No, underpowering will not "power" the device, it will not run. You must get an adapter that is 6V and (2A or higher) will be ok as well.
Yes it will
Yes, the manufacturer specifies what the working voltage of a device will be and that voltage has to be adhered to.
Most probably you are using a AC (117V or 220V) to DC 6V converter adapter. If yes, then the answer is no, you cannot use a lower current rating for a device that draws more current. If you plug in a device that needs DC 6V 500 mA to an adapter that can only supply DC 6V 300mA, then the adapter could start sending higher than 6 volts current which may damage your equipment or overheat the adapter. If you are using a higher mA rating adapter, then it's okay. For more technical details, see: http://www.jaycar.com.au/images_uploaded/plugpack.pdf
No, of course not. The 6v adapter only produces 6v, while the load expects 9v.
yes, depends on equipment..
No, twice the voltage applied to a device that only requires 6 volts will probably destroy the device. When a manufacturer of equipment states a required voltage for a device that is the voltage that must be used.
No. You need to use the exact battery charger for the battery specified by the manufacturer, in order to achieve the correct charge cutoff point. In particular NiCad and similar batteries detect full charge by detecting the knee point in voltage per unit time given a specific charge current. Using the wrong charger could result in overcharge which will damage the battery.
Yes you can. Given the fact you're talking about milliamps - the slight increase in amps will have no detrimental effect.
no
No, the charger has to be equal to or greater than the device it is charging.
A device which requires 1000mA at 9 volts in order to operate will not work from an output of 500mA. This output is only half the power requirement of the device.