Can't Load Apple Hardware Test
I would manually download the AHT for your MBA and run it from a USB. The steps to do this are as follows:


Create a bootable USB flash drive by using Disk Utility's Erase function. Make sure you choose OS X Extended for the format and the GUID Partition Map for the scheme. For the purposes of these steps, also give your USB the name AHT when you're erasing it.
Download the AHT for your particular MBA here
Mount the downloaded image (it should mount as AHTCThree)
Now you will need to copy the AHT to your USB flash drive. To start, Launch the Terminal app (usually found within the Utilities folder)
In the Terminal window, enter the following:

cp -r /Volumes/AHTCThree/System /Volumes/AHT/
Now we need to make the USB flash drive bootable, so enter the following line in Terminal:

sudo bless --folder /Volumes/AHT/ --file /Volumes/AHT/System/Library/CoreServices/.diagnostics/diags.efi --label AHT
Enter your Admin password
Leave the USB flash drive plugged in and restart your MBA
Immediately hold down the Option key
Select the AHT from the list and boot up


NOTE for other readers: - The  AHT download and Terminal commands used in this answer are only for Mid-2012 MacBook Air models (both the 11" and 13"). Do not attempt this if your model Mac is different!