I have a battery charger which says Output: 7.2V 250 mA. Can I use this charger with a 9.6V battery? If so, will the current change from 250 mA?
No, this will not charge the battery (at least not in any meaningful way) the voltage supplied to the battery must have a higher potential difference (voltage) than the battery itself or there will be no electron flow into it.
To charge a 9.6v battery, you need a charger rated for 9.6v. The current of 250ma is just the maximum safe rated capability of the charger. The battery will demand and draw what current it requires. A charger rated at a lower voltage than the battery will never fully charge the battery, and probably not even provide a sufficient partial charge. You COULD use a charger rated at a higher voltage than the battery, but you're more likely to damage the battery from overcharging it.