1) I'm a little confused about your question here, because one way or another you have to use a certain type of rechargeable battery - either just rechargeable AA's, or a rechargeable battery made specifically for the Wii Remote (http://www.ebgames.com/Catalog/ProductDetails.aspx?product_id=63662). I assume this second option is the one your looking for?
2) The wii comes with one Remote and one Nunchuck. The remote is where most of the motion control actions happen, and the nunchuck has an analogue stick on it (like on a PS2 controller), though it also has motion control functionality (Yahoo search these for more info). Most games use the Remote, some require the Nunchuck as well. Sometimes certain games (a lot of Virtual Console games) work with the classic controller, a more tradition controller (kind of like the SNES controller) which plugs into the Remote. Depending on what games you end up getting, you can determine what combinations of controllers you'd need. (You can also use Gamecube controllers for some games)
You can check what games use what controllers on the back of the game box, or with a little Yahoo searching.
3) You should be able to connect your Wii to the Internet using your wireless network without buying anything else (its just a matter of a few menu settings of your Wii)
4) If you liked Super Mario World, then Mario Galaxy is a cool transition into the next generation of video games. For sports games, the MLB Power Pros series is really great (I don't like Pro baseball all that much, but I really enjoyed last years version, and I think the 2008 version is due out soon). I think most of the EA games for Wii are pretty weak so far, so I couldn't recommend any (even though college football will be available on Wii this year). I dunno if you enjoy soccer, but Pro Evolution Soccer 2008 is a fantastic game that really takes advantage of the Wii's functionality.
And I dunno if you like racing games, but I'll also include Mario Kart Wii, just because its one of my favorite games to play with other people...
(and then of course, theres all the Virtual Console games you can download, including Super Mario World, also one of my favorites!)