If the software was completely resinstalled ( is it Lion?) then it does sound awfully like a hardware problem, maybe the video card. The thing about all Macs, and especially the laptops, is that they are not designed to be repaired. I have thrown, literally, so many away as the price of fix is so high.
The fact that the Air has no dvd/cd drive makes it very hard to run freestanding disk utilities on it. If it has Firewire you could link it to another mac and run the utilities from there after starting the airbook in target mode. You may be able to do something similar with the wireless, but I dont have an Air myself to know if thats possible.
Of all the disk utilities I have found DiskWarrior the best of all. It's not cheap tho. Apples own disk utility is really not much use 9 times out of 10 and is running off the hard drive its trying to fix, so logically it hasnt really got a chance of fixing deep problems
I presume you are out of warranty? In which case best scenario may be to get all your data off the air via wireless network and prepare to throw the machine away. Second hand is cheaper than a repair.
Or do as I did and go darkside for laptop. I got an acer with windows 7 for £399 and while it is no beauty to look at, it runs very well and very fast and very reliably and win 7 is actually not bad at all.
I cant see why the kindle would have done any harm except you must unplug any usb devices on start up in case the Air is trying to mistakenly boot up from the external device