How do I keep the Recovery HD and EFI partitions from automatically mounting on my MacBook Air?
They are hidden partitions in OS X. Showing all partitions in Disk Utility also shows the partitions in the list:



…and diskutil list:

1:         EFI EFI          209.7 MB   disk0s1
2:   Apple_HFS Mac SSD      150.0 GB   disk0s2
3:  Apple_Boot Recovery HD  650.0 MB   disk0s3


They are usually not mounted by default and therefore not shown on the desktop regardless of your settings in Finder.

Try resetting your NVRAM (boot with ⌘⌥PR until the second boot chime).