broken EFI / missing space after re-partitioning — using mavericks 10.9.5
THere's nothing actually wrong with your Air's EFI, what you are experiencing are the death-throws of a failing-and-fading-fast SSD module.

We tend to be rather blasé about solid-state drives these days, but seven years ago the technology was still maturing, and SSDs were still susceptible to sudden and inexplicable demise. When Jobs decided to start fitting SSDs into the first MacBook Airs, Apple did a lot of destructive data testing until the found a brand that they thought might last five years - Toshiba.

I'd look at getting a replacement SSD-stick - Transcend make a nice JetDrive replacement for your machine.