The half life of a 1st order reaction is constantly decreasing because the rate of the reaction only depends on one of the reaction components. When that component is half gone, the reaction rate is halved by the same amount.
The half life is always constant for a first order reaction. For a first ordered reaction the half life time is .6930/rate constant and the rate constant doesn't change... it's a constant. Yes, you're right the reaction rate goes down, but that's because the concentration of [A] is decreasing with time. It doesn't affect the half life. This ones more of looking at the equation.