Yes
The battery inside the mouse will need periodically charging.
bluetooth mouse
Make sure the mouse is plugged in and or/ has a working battery if it is wireless, if nothing works get a new one.
The mouse will worse on practically any service as long as there is sufficient battery power.
There is no way. If you want a wireless mouse you will have to purchase one.
Wireless gaming mouse
If it is wireless, check the battery, but if its on a cord check for frayed wire and make sure the ball is in place. if the ball isn't in place put it back in correctly, if it wont spin,WD-40, in small amounts MAY help
A wireless mouse can be used for both. As for recharging, most wireless mice you will look at take Alkaline batteries, so you have to replace them when they die. If the mouse contains Rechargable batteries, then yes it will have to be plugged into the PC (or wall) to charge.
Wireless mice have the advantage of not having any cables.
a mouse
I would recommends the Wireless Mighty Mouse by Mac.