Trackpad tap to click malfunction
I suggest you book an appointment with an Apple genius.

I have had a Magic trackpad for my iMac for a couple of years.

But stated to get the same issues you have around the time I installed 10.8.5 ( or one of the later updates )
 An additional issues was in any app that text could be selected the track pad would do so randomly and of it's own accord.

I thought it could be be the software and tried all manner of things.
Killing the plist included.

But the problem persisted into my install of 10.9.

I finally took it to a genius two weeks ago who agreed the issue was most likely hardware.

They replaced the Magic trackpad with a new one and I have not had any problems since.

So I think to save yourself some time and frustration book an appointment.
Also  because it is a trackpad on a MacBook Air  they should be able to run proper diagnostics and they should be able to sort it one way or other.