How can I tell what version of Recovery HD OS is running while I'm booted to the recovery software? How can I tell the version of OS X (and the build) that runs on Recovery HD while I'm running in that mode? The Apple menu has no "About" menu that shows that information when you're running normally.
The easiest command is to use Terminal - it's under the Utilities menu - and type the sw_vers command:

sw_vers


The output will be the version and build that runs when you boot to Recovery which can be different than the version installed on your main bootable OS X partition.