My first question is, how would I go about reducing down the thickness of this handle? My second question is, if I reduce the handle down, my brace height will change but will my draw weight change? Thank you so much, John.    

If your reducing from the front where your hand rests.I would start with a rasp or a drum sander and shape it as you want it.
Then sand smooth.

If that is the area you reduce then yesmthe brace height would measure different, but this can be changed by adjusting string length.

If anything given the same string the weight might be reduced a pound or two. Nothing to be concerned with.

How is the draw weight going to change by making the grip smaller?

The draw weight will not changes at all unless you remove material from the limbs. Just altering the grip won't make a bit of difference to anything other than the shape of your grip (unless of course you changed the location of the pivot point).
Don't adjust the string and you won't have affected anything. Your arrows will still be fine and will hit where they were hitting before.

I think he might mean reducing the depth of the grip which would reduce his draw length slightly.  Same as changing the pivot point.

Draw weight is measured as the weight pulled at a given distance in relation to the back of the bow.  This would not change by making the grip area smaller (i.e. less deep).

As thick as that handle leather is i'd take that off and see if that does enough. A thin piece of leather would change the feel a lot.