How do I show that the total system has the impulse response $h[n] = aδ[n] + bδ [n - 1] + cδ [n - 2]$ and determine the constants a,b and c?
Since this is a homework type problem I will not present the complete solution, but I'll help you to solve it yourself.

First of all, don't write out the convolution as an infinite sum, but simply use the fact that for any sequence $x[n]$ the following holds:

$$x[n]\star\delta[n-k]=x[n-k]\tag{1}$$

The lower path of the given system has the impulse response

$$h_2[n]\star(h_3[n]+h_4[n])=h_2[n]\star h_3[n]+h_2[n]\star h_4[n]=h_a[n]+h_b[n]\tag{2}$$

and if you use $(1)$ to evaluate $h_a[n]$ and $h_b[n]$ in $(2)$ you'll quickly see that both are (very simple) finite impulse response.