Using the attachment as an example, if I were to leave pins 1 and 3 disconnected, but connect a speaker through pin 2 and a light through pin 4, would this be a viable method of doing them both at the same time? I have a piece of audio that needs to play over several speakers (never at the same time though) and also need a light to come on corresponding to the speaker playing. Are there any major issues with doing this or thing I would need to be aware of? Would it be better to use separate relays? Lastly, should I have separate grounds for the sound and light circuits or does it not matter?
Yes. The two sets of contacts are isolated from each other (if you wire it correctly).
but connect a speaker through pin 2 and a light through pin 4
The speaker signal would "go through" T1 & P2, and the light power would "go through" T2 & P4.
Just be careful and don't mix-up the connections. Double-check with an Ohmmeter before connecting the speaker or light. Especially, make sure there is no connection between the light and speakers circuits (with the relay energized and with the relay not energized).