Yes, yes you can. And the beauty is that I can tell without even looking at the Si4735 shield you are using.
How is this possible you ask? The only pins that the AudioShield use are the SPI lines (MOSI,MISO,CLK), a SPI chip select line, and an analog pin for reading the potentiometer. Obviously, the A3 analog pin for the potentiometer can be moved around and put on any other analog pin. The chip select is just an active low digital signal used to determine which SPI device should currently be using the data on the SPI lines (the SPI protocol was designed for a shared bus).
So all in all what that means is for the AudioShield, all of the pins it uses were either designed to be shared, or are not really pin specific and through some super-simple hacks can be moved around. The link you provided actually explains how to move the chip select signal that I talked about to a different pin and the potentiometer hack is basically the same but you have to move it to an analog pin. Of course any code samples you use will have to be adjusted for the new pins you use.
For more information on the arduino pins see here.
For more information on sharing the SPI bus between two devices see here.
For reasons I pointed out in Transmitting a video stream through a microcontroller's wifi, Arduinos are not suited for audio/video streaming. The audio portion of this makes this even harder since the Arduino would need to combine the video with the audio. In fact, "not suited" is being polite, I doubt the Arduino is even able to handle this.
A 32bit MCU or DSP is far better suited for this!
How to send camera video feed and microphone audio feed to the shield
(WiFi or Ethernet)?
Most cameras for this type of application transmit their video via a serial interface. These are not advanced cameras (since the Arduino is so under-powered.) These camera's usually encode/compress video as Motion Jpeg (MJPEG), which is then sent via serial communication.
I can't seem to find one, but I would think that there are camera's that include a mic and encode the video and audio together and transmit it via serial so you would just need to connect it to one of the Arduino's serial ports (or just bypass the Arduino and connect it straight to a serial to WiFi/Ethernet processor like the shield you mentioned.)
If the camera does not handle the audio side, I suppose you could sample the audio using the Arduino's ADC (not recommended) or use some other processor (Codec Shield) to encode the audio (much better quality) and send it to the Arduino. Then the Arduino would need to encode the audio with the video (something I highly doubt it is capable of) and transmit it to the WiFi/Ethernet board via another serial port.
How does the shield actually transmit the A/V feed? Does it
automatically encode the stream or is this something I'd have to
program myself?
Think of the shield as just a serial cable, all it is doing is transmitting the serial data it is receiving. So however the audio and video is encoded, it will need to be decoded on the other end (a simple task for a computer.) An example of this for the video is MJPEG, the camera encodes/compresses its raw pixel data into JPEG format and transmits it. The receiving device decodes/un-compresses the data and displays it.
Helpful links:
Best Answer
You can use a VS1063 to do on-the-fly conversion of audio to MP3. VLSI offer an eval board that you can hook up to a microcontroller, or even a toyduino, to configure and stream from the VS1063. The pro kit includes all the audio circuitry and connectors.