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7.4V is a double cell. A single cell would only have 3.7V and not be sufficient for your drone. Additionally, Li-Ion Batteries do not provide enough current to drive a drone. You will need a Li-Po battery for that.So a single cell Li-Ion would not have sufficient current, not sufficient voltage and not be the right battery technology for what you are planning.
My company now does not have this rule anymore but they did have it a few years back because they use the 2.4GHz band that is also used by Wifi and Bluetooth. In an office environment, this can be a lot of devices using the same frequencies and they can cause problems with some channels of Wifi. One mouse isn't a problem, but 1000 mice in one building can be. With the switch to 5.8GHz Wifi, this problem is not a problem anymore.
FatShark FPV Goggle Replacement BatteryView Instructable »
Lithium Field Charger Battery PackView Instructable »
Of course it is. And swapping a battery is done in like half a minute. But if your office has a strict "no wireless mice, they affect our wifi" rule, you're out of luck. This instructable, some EMI protection spray on the inside and you're good. And there actually are people who hate swapping batteries and prefer a wire. The mouse also gets lighter without the battery. ;)
This mod is very generic. If the USB receiver fits somewhere into the mouse and the voltage used by the mouse is below 3.5V (because of the approx. 1,5V voltage drop of the LM317 relative to the USB voltage), then yes, it will work. The Performance Mouse MX uses a single 1.5V battery, a mouse using two of them would be at 3V and still possible with this mod (but with different resistors connected to the LM317, google "LM317 calculator").
First of all, I'd replace the battery with a new one. I know it is rechargeable but those have a limited age, too. If the lights on the mouse (battery indicator) do light up and you still can't pair your mouse with your dongle, try another computer to make sure it isn't the driver. If that also fails, I'd get another dongle (from a friend or colleague) to try. And if that one also fails, the mouse is dead, time for a new one.I don't think you can repair a dead mouse if it wasn't you who broke it and the broken part is clearly identified, can be bought in quantities less than 1000 pieces and is replaceable.