MacBook Air (1st gen) running really slow, basic apps crash and no internet
Have you run the Apple Hardware Test that came with the MacBook Air?


  To use Apple Hardware Test on your
  MacBook Air:
  
  
  Disconnect all external devices from your computer except the power
  adapter.
  Restart your MacBook Air while holding down the "D" key.
  When the Apple Hardware Test chooser screen appears, select the
  language for your location.
  Press the Return key or click the right arrow button.
  When the Apple Hardware Test main screen appears (after about 45
  seconds), follow the onscreen
  instructions.
  If Apple Hardware Test detects an issue, it displays an error code. Make
  a note of the error code before
  pursuing support options. If Apple
  Hardware Test doesn't detect a
  hardware issue, the issue may be
  software related.
  


If AHT found no issues, run a memory test. Download memtest86+ and burn that to a CD, and boot the MacBook Air from the CD. I have personally had RAM issues which AHT didn't find but memtest did.

EDIT: Sorry, I only just not noticed that you said you're unable to run AHT, presumably because you don't have an external optical drive. I'd recommend taking the MacBook Air in to an Apple store. Or by using a USB Ethernet Adapter you could probably get it to boot AHT from the network.

EDIT2: Have you tried booting from a USB drive? You could use another Mac to install a fresh copy of OS X to a US disk and see if the Air can boot that. or, you might be able to copy the Mac OS X Install DVD to a USB disk and try to run AHT off there...

Ultimately, a visit to the Apple Store is probably your best bet :-(