Hartree-Fock correction to $e$-$e$ interaction The corrections to the energy per electron in a jellium model (uniform distribution of positive ion charge approximation to the regulated long range order ionic array) is given by (in units of Ry) 
$$\frac{E}{N} = \frac{2.21}{(r_s/a_o)^2}  − \frac{0.916}{(r_s/a_o)} +0.0622\ln(r_s/a_o)−0.096+ O(r_s /a_o ). $$
($r_s$ is the radius of a sphere that holds the same volume as one conduction electron and $a_o$ is the Bohr radius, but I don't think is so important to my questions). See e.g. the derivation and this same result in Ashcroft and Mermin P.336.


*

*My question is why does the last three terms contain contributions to the kinetic and potential energy? It seems to me that it should only be potential energy since the source of the extra terms comes from the perturbation which is precisely due to the e-e interactions, but I have been told otherwise.

*My other question is this whole procedure was implemented to better account for the interaction between the electrons in a metal. This expansion above is only valid for $r_s /a_o \ll 1$ and for most metals this value is from 2 to 6. So what is the usefulness of this result then if it does not apply to the very thing it sought out to describe? 
 A: Even if it's not really appropriate to derivate quantitative results about electrons properties in solids, Jellium model still have some interesting qualitative features. As you correctly pointed, Jellium still a mean-field theory and so fails about dealing with strongly correlated systems.
Lets remind a few about jellium model. The starting point is the hamiltonian :
$$
\mathcal{H}=\mathcal{H}_{e^-}+\mathcal{H}_{i^+}+\mathcal{H}_{e^-/e^-}+\mathcal{H}_{e^-/i^+}
$$
where $\mathcal{H}_{e^-}$ is the free electrons kinetic energy term, $\mathcal{H}_{i^+}$ is the positive electrostatic ionic background, $\mathcal{H}_{e^-/e^-}$ the electron-electron repulsion term and $\mathcal{H}_{e^-/i^+}$ the interaction between the electrons and ionic background.
By introducing a scale $\gamma$ at which Coulomb interactions between electrons are consistant, which consists basically in taking :
$$
\mathcal{H}_{e^-/e^-}=\lim_{\gamma\rightarrow 0}\frac{e^2}{2}\sum_{i\neq j}\frac{e^{-\gamma|\textbf{x}_i-\textbf{x}_j|}}{|\textbf{x}_i-\textbf{x}_j|}\equiv\lim_{\gamma\rightarrow 0}\frac{1}{2\Omega}\sum_{\textbf{k},\textbf{p},\textbf{q}}\frac{4\pi e^2}{q^2+\gamma^2}a^\dagger_{\textbf{k}-\textbf{q}}a^\dagger_{\textbf{p}-\textbf{q}}a_{\textbf{p}}a_{\textbf{k}}
$$
Such procedure allows us to get rid of divergences introduced by the $1/r$ decreasing Coulomb interaction. And interestingly, it can be shown that the $\mathcal{H}_{i^+}$ background energy is compensated by the $\textbf{q}=0$ term of $\mathcal{H}_{e^-/e^-}$ at the thermodynamic limit ($\Omega\rightarrow+\infty$, where $\Omega$ is the volume of the system). Then, one can end up with :
$$
\mathcal{H}=\mathcal{H}_{e-}+\mathcal{V}=\sum_{\textbf{k}}\frac{\hbar^2\textbf{k}^2}{2m}a^\dagger_{\textbf{k}}a_{\textbf{k}}+\frac{e^2}{2\Omega}\sum_{\textbf{k},\textbf{p},\textbf{q}\neq 0}\frac{4\pi}{q^2}a^\dagger_{\textbf{k}-\textbf{q}}a^\dagger_{\textbf{p}-\textbf{q}}a_{\textbf{p}}a_{\textbf{k}}
$$
Regarding your first question, it is now clear the $\mathcal{V}$ term is the result of the "mixing" (said mutual counterbalance) of various contributions of the initial hamiltonian, namely :
$$
\mathcal{V}=\lim_{\Omega\rightarrow+\infty}\lim_{\gamma\rightarrow 0}\,\mathcal{H}_{i^+}+\mathcal{H}_{e^-/i^+}+\frac{1}{2\Omega}\sum_{\textbf{k},\textbf{p},\textbf{q}}\frac{4\pi e^2}{\gamma^2+q^2}a^\dagger_{\textbf{k}}a^\dagger_{\textbf{p}}a_{\textbf{p}}a_{\textbf{k}}
$$
To obtain the $E/N$ OP's expression, one has just to compute $\frac{\langle F|\mathcal{H}|F\rangle}{N}$ by using a perturbation theory, where $|F\rangle$ is the Fermi sea :
$$
E=E^{(0)}+E^{(1)}+E^{(2)}+...
$$
Such perturbative expansion is valid only if Coulomb interaction is negligible compared to kinetic energy. Typical kinetic energy scales like :
$$
E_{\text{kin}}\sim E_F\sim n^{2/3}\sim 1/r_s^2\quad\text{where}\;r_s=1/n^3\;\text{with}\;n\;\text{is the density}
$$
$r_s$ can be interpreted as the typical distance between two electrons.
Typical Coulomb energy scales like :
$$
E_{\text{Coulomb}}\sim\frac{e^2}{r_s}\sim\frac{1}{r_s}
$$
Then, it should verify :
$$
\frac{E_{\text{Coulomb}}}{E_{\text{kin}}}\sim\frac{1/r_s}{1/r_s^2}\sim r_s<<1
$$
Regarding your second question, let just plot $E/N$ at first order of perturbation :

As you can see, it exhibits a negative minimum around $\frac{r_s}{a_0}=4.83$. This negative energy minimum actually refers to metallic bonding (attractive interaction), where electrons are somewhat "gluing" ions between them.
