Update batteries and buy a new one
No, the wireless mouse adapter was meant to receive the signal from the mouse only. This is because there are many wireless devices and your adapter was designed only to pick up the one registered device. Even if you could donate the mouse responder to the router, the adapter would expect the router to send it mouse specific commands only.
no, wireless mouses and keyboards come with usb plugins so that it is wireless, so therefore, you can have a wireless mouse without a wireless keyboard.
Buy a new one.
Physically/hardware fix it? or fix the settings? MOAR INFO needed.
A usb wireless adapter is useful to obtain a wireless connection on older computers or when an internal wireless card is nonoperational. It can be an easy fix instead of having to take a part ones computer.
Sometimes signals from different wireless devices might conflict. But usually a wireless mouse cannot confilct with a wireless network adapter (however, it's possible). You have read the manual for the mouse and check on which wavelength it operates. After you have read the manual you will know the frequency or frequency range and it will be easier to understand where you can get problems.
Buy a new one.
reset it and try again or contact Microsoft
Yes, as long as you have the usb ports for the adapters. Also, there are some brands that sell a combo of the keyboard and the mouse together that will use the same adapter if you do not have very many usb ports.
Xbox Wireless Network Adapter MN740
you buy the wireless adapter
Extended service set (ESS):Also known as infrastructure,where hosts connects to a wireless access point using a wireless adapter.