In the image above the second faint part is the answer to the question c). The answer starts with the word evidently, and I am confused why it is evident. Also, since we are considering a shell, shouldn't it be a 3D form of Delta Dirac function? Is there anything wrong with the answer?
It is evident because there is no charge anywhere except $r=R$, therefore the charge density will have a $\delta(r-R)$ to capture this. ... and it is indeed $\delta(r-R)$ because this is spherical distribution which depend on the radial coordinate only, rather than all $3$ coordinates, i.e. you are describing a surface (2d) of fixed radius which does not depend on $\theta$ and $\phi$.