Changing the power input to the LED will definitely change the brightnes of the LED. I have run my Luxeon V LED's from 4.8 watts up to around 10.8 watts and there is a huge difference in brightness as I vary the power input.

Yes, voltage and current are relational just like Novastar said.

Yes the equation for power dissipation is P=I*V, it's also... P=R*I^2

Yes an increase in voltage on your LED results in more current on the LED, which results in more power being used, which results in more lumens at the LED.

The best way to see minior increases in brightness is two identical sabers side by side... but different values of current. Looking at one saber by itself it's really hard to see differences in power usage.