Yes to the first and why bother to the second.
You can charge the battery outside the camera using a BC-10L charger or equivalent. One such charger is available here
The best way to download pictures from your camera to your computer involves removing the memory card from the camera and plugging it into a card reader (either built-in to the computer or connected via USB or FireWire). This is likely to be faster than connecting the camera to the computer, and won't run down your camera's batteries.
Once the card is plugged in, it will appear to your computer as a removable drive. You can use the operating system's drag&drop facility to copy pictures from the card to the computer's hard drive, the same way you copy any other files. Or you can use any photo cataloging program such as Picasa