Diode reverse current
If I apply an current source into the the cathode of a diode (current
  flow into cathode), what can possibly happen to the diode.


Which diode?  


The desaturation check diode
The schotkey diode at the power supply
The diode setting the the turn-off characteristic


I am going to assume you means #1 as the other's are not connected to anything and this appears to be part of either a phase leg or some pulled down load. In future please be concise...

Depends. Is this a theoretical case such that "ideal current sources" exist then the voltage associated with such an ideal current source could increase to infinite voltage to ensure the constant current would flow. This would cause a real diode to avalanche and be damage. But if we are contemplating ideal components then the diode could be ideal and thus could block infinite voltage and equally the FET could block infinite voltage. 

If you are discussing a hypothetical load at the Drain of the FET then as long as the blocking voltage of the diode (and the FET) are higher than the forcing voltage behind whatever can source current... nothing will happen, at this part of the circuit. there will be some leakage current flowing, the value of which depends on the specific device. 


  For the cuicuit below, when the FET is OFF, how does the battery
  current flow. Will the current flow through the diode while it's in
  reverse region.


What battery? there is no battery.