How does the factor of dimensionality arise in the derivation of the diffusion equation?

Question

I am trying to understand the derivation of the diffusion equation. I am using the following document:

The particular derivation is here:

I am stuck at two problems, that I cannot understand

1. Why is $ <r_i r_j > 0$ when $i \neq j$
2. I don't understand the factor $1/d$ arising in the last term.

I suspect I have to use the property:

$$ \int r^2 p(r) dr = a^2 $$