How can I dampen current spikes when using NPNs to drive an LED
You're seeing current through the parasitic capacitance of the diode. 

You could avoid it by turning on the transistor more slowly.

But you shouldn't really expect this simulation to be predictive of what you'll see in a real circuit, since your LED is not likely to have same parasitic capacitance as a 1N914. Also the power source you use in the real world will have parasitic resistance and inductance that will reduce the overshoot current.