Nintendo has discontinued the USB Wi-Fi Connector, so definitely look into a wireless router if possible. The people I know who did get one had some issues setting it up, but when they did it worked ok. They did mention that it gets hot, though.

Separately, there's a 'Datel WiFi Max' that apparently works in a similar manner.

The way they work is that they plug into a computer that is connected to the internet, then broadcast a wifi signal for the DS to pick up. Much more convenient then a wireless router if you move around a lot, though I think a router would be more effective overall.

I'm not 100% on this one, but I heard the Wii (connected via ethernet) can be used as a WiFi access point for the DS if a WiFi network doesn't exist. I'm unable to find any confirmation of that, so don't invest too much thought into it. Particularly if you don't have a Wii

I think the DSes come with a decent charge when you get it. I highly doubt you have done any damage to the system whatsoever.