I have to let you know that these aren't strictly legal, so if a moderator or admin thinks this is inappropriate, please feel free to remove this information.
1. They mimic the running of a games console, effectively running a virtual version of them on your computer. They utilise various libraries and assemblers to allow the game to communicate with your computer's processor.
2. Many emulators have recorders built in to them. Otherwise, you can use FRAPS or another such recording/screeen capture tool.
3. There may be utilities which allow you to use your PC's wifi or internet connection to interface with another emulator, or even a DS. Google it.
4. From my research, in order, the best are: NO$GBA, DeSmuME, iDeas, and NeonDS. Search them. And to prevent getting a virus, make sure you scan the downloaded file before opening it, or use firefox.