Hello, my name is Dan.
I will do my best to help you, the only stupid question is the unasked one.
Good diagnostics, do you have another Mac computer with an optical drive or an external optical drive (DVD drive)?
I ask because it will be very handy for what follows.
I use a utility that picks up where Disk Utility leaves off called Disk Warrior, it can repair drives that DU cannot.
Another one that works well is TechTool Pro
Both can boot your Mac and are about $100 each and worth every penny.
The reason for the DVD is to boot from it to do repairs, the MBA is started in Target mode, connected to the other Mac via Firewire or, in some cases, USB.
The utility is used to either repair the disk or the host Mac can be uses to install OSX on the MBA. The MBA is treated as an external hard drive.
Once that is done backing up is paramount because it may not hold.
That is a partial answer, please look it over and let me know.