How does the emulator get connected to the internet?
The emulator is a program on your computer like any other. The program can connect to the Internet in just the same way Firefox can. In particular, this means it makes no difference what kind of connection the host is using.

From inside the emulator, there's a 'fake' network driver. It appears to Android like a mobile Internet connection, but instead of communicating with a cellular modem, it's actually using the usual network API of your host OS to send and receive data, just like Firefox. It also does emulator-specific things like simulating packet loss.