After looking around, I would say that the answer is yes. However, you might get a better deal by comparing the prices from several different dealers.
Wii remotes need two double A batteries to work.
No, the Wii Nunchuk doesn't require batteries. It is wired and is connected to the Wiimote. However you do need batteries for the Wiimote itself to work. There are some wireless nunchuks from third party companies that require batteries though.
No, it uses Wii remotes (that need batteries) but I think the microphone plugs into the back of the Wii.
AA
2 of them
Mine I have to replace with batteries.
yes they go in your wii remote to plug into the guitar controller (you need AA)
Wii remotes run on 2 AA batteries but if you can get a pair of recharable batteries then yes you would need to charge them. (Pick your brand carefully, some batteries run out fast!)
because you need to have batteries in the remote and the wii has to be plugged in and on
they take the same as the wii controllers (AA) because you need to plug in the wii remote into the guitar controller to enable it to work.
You don't have to buy one. The Wii Remote is designed to use two AA size batteries. The charger packs are a convenience, as you don't have to go out and get new batteries every few hours of gameplay.
WII remotes are electrical devices which consume power. All devices which use energy require a source for this energy. Wii decided that batteries made the most sense for the energy source.