Discrete Filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$
You can see by definition that your system is LTI System.
Moreover it is casual as it is only depends on input from the past.  

The quaestion asks for response on the Unit Pulse signal on time 0.
In LTI Casual System it is the coefficient which responds to to input at time $ n $ -> Its value is $ \frac{1}{3} $.

I'm not sure where $ h \left[ 1 \right] + h \left[ 5 \right] $ is from though.
As clearly this system can be represented by 3 Coefficients which you found in Z Domain (Though they are also given from the finite differences equation).