How to apply the Barkhausen criterion in order to know if a system will oscillate?
The Barkhausen criteria are usually applied to analyze sine wave type oscillator circuits (Wien bridge, etc.) where a small signal, such as thermal noise, is exponentially amplified around the positive feedback loop to create the output signal.  Oscillation is inherently a large signal phenomena and in general can't be analyzed using LTI analysis methods, but the Barkhausen criteria let you predict oscillation from the small signal gain and phase behavior.

It's less clear to me how to directly apply such techniques to this relaxation oscillator circuit, as circuits like this don't have any small signal behavior - there are only 2 stable states.  It should be fairly obvious, however, that whatever component values you choose the feedback around the loop will eventually be unity and in phase, i.e. when V+ = V-.