Unable to boot into recovery mode on El Capitan MacBook Air
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.