huge reverse current leakage in fast recovery diode for boost converter
The 10ns spikes are given by parasitic oscillation with L or transistor pins parasitic inductance and diode capacitance when switching while L is still charged, you can notice that if you show the current through one inductance on the graphic.  

Make some soft start like use variable frequency first then fixed to avoid that

Update 

If you look at the drain current you can notice the same negative current glitch while turning on meaning of that is that it's parasitic inductance is causing that. 

Same solution.  

Zoomed graphic



The positive glitch is the drain current , the negative is the diode current , down, in purple, the gate voltage. The zoom is in the region where M1 is switched on while L still charged

1) and 2) Nothing to do with the surge current , a real diode has a parasitic capacitance,  the current is going through it no matter the breakdown voltage. Together with the drain parasitic inductance it makes a resonant circuit that makes the current even higher.