= Solution
Normalize the present <scale factor> to $a_0=1$. Conservation of the number of electrons associated with the baryons makes their physical <number density> scale as $a^{-3}$. At this early epoch they are free rather than bound in neutral atoms, so
$$
\boxed{
n_e=n_{e,0}\left(\frac{a_0}{a}\right)^3
=0.2(10^6)^3
=2\times10^{17}\,\mathrm{m^{-3}}.
}
$$
Here the specified present density is interpreted as the conserved total electron abundance, including electrons that might be bound today. A present free-electron density would require an ionization-fraction correction.
Density alone does not determine whether a gas is relativistic; its random kinetic energy must be compared with the rest energy. In the usual thermal <Hot Big Bang model>, use the measured <cosmic microwave background> temperature $T_0\simeq2.7\,\mathrm K$ and the <temperature scaling of thermal photon number and energy densities>, $T_\gamma\propto a^{-1}$. Tight coupling then gives
$$
T_e\simeq T_\gamma\simeq2.7\times10^6\,\mathrm K,\qquad
k_BT_e\simeq230\,\mathrm{eV}.
$$
Since $m_ec^2\simeq5.1\times10^5\,\mathrm{eV}$,
$$
\boxed{
\frac{k_BT_e}{m_ec^2}\simeq4.5\times10^{-4}\ll1:
\quad\text{the electrons are nonrelativistic}.
}
$$
Their thermal speed is of order $\sqrt{3k_BT_e/m_e}\simeq0.04c$. The gas is fully ionized because its temperature is far above the <recombination temperature>, but ionization does not imply relativistic electrons. Thermal electron-positron pairs are exponentially suppressed at this temperature.<Cosmology by David Tong>[Section 2, thermal history and recombination: present photon temperature, electron mass scale, and temperature-redshift scaling]{ref}
The given age and scale-factor ratio should be treated as independent supplied estimates in the following rate comparison. They do not describe the same epoch in standard radiation-era expansion: $a/a_0=10^{-6}$ normally corresponds to an age of order a year, not $10^4$ years. The density and temperature estimates above depend on the scale-factor ratio, not on assigning it the stated age.
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