Question 2 of the 2013 paper is a bit unusual, mostly because we are dealing with a long range interaction for the first time.
What I should like to know, is why 'The electrostatic energy due to the
interaction of the electrons with the ion cores and among themselves' does not depend on the electron charge or density. This is most baffling, since we are asked to work out their kinetic energy as a function of density, then minimise the sum to find the density that minimises the interaction energies. Unless both terms in the sum depend on the density, the result will be boring indeed! On physical grounds the electrostatic interaction term must depend on the electron density No electrons, no interaction!