Q: iMac bluetooth keyboard is muting the sound without the F10 key being pressed

Restart the computer holding down the command - option/alt - P - R keys before the gray screen appears. Continue to hold until you hear the startup chime for the second time. It won't hurt anything.

Reset NVRAM/PRAM

From the article link above.

A small amount of your computer’s memory, called “parameter random-access memory” or PRAM, stores certain settings in a location that OS X can access quickly. The particular settings that are stored depend on your type of Mac and the types of devices connected to it. The settings include your designated startup disk, display resolution, speaker volume, and other information.