Moving an object in the a particular view should never move that object in the axis aligned with that view, ie, moving it in the top view moves it in the XY plane, and leaves the Z position alone.
However, your current 0.03 set up has it snap to the nearest gridpoint in all three axes when moved.
I'd also prefer if it didn't snap to the centre point for brushes. It's frustrating to make your 16 unit thick brush, then move it only to have it snap to the centre. It means you either have to continually work on one grid size lower than you want to or you have to continually move the sides rather than the brush as a whole.