Unable to boot into recovery mode on El Capitan MacBook Air
Solution 1:
You could do Command+Option+R to boot into internet recovery (from Apple servers). Ensure a stable internet connection. You can then run csrutil disable
in the terminal.
As far as your recovery partition. If it is truly not working, you can delete the partition, download the latest OS X installer from the App Store and then reinstall OS X (this will not delete your files, think of it just like an OS X update). As part of the OS X reinstall, it will add a fresh new recovery partition.
Hope that helps.
Solution 2:
Solved (for my particular situation anyway): I was suspicious that my machine (2009 Mac Pro) wouldn't recognize the keyboard through an external monitor. So I pulled the USB from my monitor, plugged it directly into the desktop, and lo-and-behold, Command+r allowed the machine to boot into recovery mode.