OK, at this point with the multitude of glitches you're seeing, I think an OS reinstall is the cleanest way to get a fresh start.
With your MBA you got two disks, one of them is "Disk 1", the install disk. Assuming you have the MBA drive, insert that into the drive and reboot, holding the "C" key down as soon as you hear the chime. The Mac will present the HDD and the Install DVD as boot devices.
Select the Install DVD and when prompted to install, you'll have three choices: Clean Install, Archive & Install, and Upgrade. You want to select Archive & Install. This will leave all your user settings, applications, and documents as they were, and install a fresh System folder and move the old system into a Previous System folder. The Previous System folder can normally be discarded safely.
If you can't install for whatever reason (don't have an MBA drive, for example), or the reinstall does not correct your problems, get thee to an Apple Store. The MBA is certainly under warranty, and this is something that Apple should be looking into. Other possible causes for these issues are a dodgy logic board, bad RAM, or failing HDD, in decreasing likelihood.