What was the pressure of the Universe, when the CMB was 300K? I suspect, at the time when stars didn't exist yet, we could calculate with largely homogeneous Hydrogen-Helium mix.
Unfortunately, the sources I've found on google aren't really detailed.
 A: Very, very small.
Today, the temperature of the Universe is $T_0 = 2.73\,\mathrm{K}$, so when the temperature was $T = 300\,\mathrm{K}$, the redshift was
$$
z = \frac{T}{T_0} - 1 \simeq 100,
$$
which means that the Universe was only $t = 15\,\mathrm{Myr}$ old, long before the first stars were born. However, this was still long after the Universe went from being radiation-dominated to being matter-dominated, which happened roughly at $t = 50\,000\,\mathrm{yr}$. For radiation, the pressure is $p = u/3$, where $u$ is the energy density, but in a matter-dominated the pressure is negligible wrt. the dynamics of the Universe.
The density of baryonic matter (dark matter is pressureless) today$^\dagger$ is $\rho_\mathrm{b,0} = 4\times10^{-31}\,\mathrm{g}\,\mathrm{cm}^{-3}$, so at $z=100$, when the scale factor (the "size" of the Universe relative to today) was $a = 1/(1+z) = 0.01$, the density was
$$
\rho_\mathrm{b} = \frac{\rho_\mathrm{b,0}}{a^3} = 5.5\times10^{-25
}\,\mathrm{g}\,\mathrm{cm}^{-3}.
$$
As you say, you can assume a gas consisting virtually of a homogeneous mix of hydrogen and helium. With a mass fraction of the two of $\{X,Y\} = \{0.76,0.24\}$, the "mean molecular weight" (average mass in terms of the hydrogen mass $m_\mathrm{H}$) of the (fully neutral) atoms is
$$
\mu = \frac{1}{X + Y/4} = 1.22.
$$
Thus, the particle number density was $n = \rho_\mathrm{b}/\mu m_\mathrm{H} \simeq 0.3\,\mathrm{cm}^{-3}$, and the pressure was
$$
p = n k_\mathrm{B} T \simeq 10^{-13}\,\mathrm{Pa}.
$$

$^\dagger$Let me know if you need details on how to calculate this.
A: The CMB has a temperature of $T=2.73(1+z)$. So when it was at 300K, the redshift $z=110$. At this redshift there were no stars (or at least for the purposes of this question, there is no significant reionisation, which happened at $z\leq 20$, so we can assume the universe is neutral H and He atoms.
The universe at that time would have been matter dominated (the transition between radiation dominated and matter dominated occurred at $z\sim 3500$, so radiation pressure is negligible). The density therefore falls towards the present day as $(1+z)^3$ - i.e. the density then was $1.37\times10^6$ times higher than the matter density today.
Assuming that by pressure you mean the normal gas pressure contributed by baryonic matter, then you could assume that the universe consists of 92% H and 8% He atoms by number and that the present day baryonic matter density is given by $\Omega_b \simeq 0.044$ (WMAP) multiplied by a critical density of $8 \times 10^{-27}$ kg/m$^3$, and with a number of mass units per particle of $\mu = 0.92\times 1 + 0.08\times 4 = 1.24$.
From this we obtain the baryonic density of the universe at $z=110$ as $1.1\times10^{-20}$ kg/m$^3$ and then using
$$P = \rho k T/\mu m_u$$
we get $P = 2.2\times 10^{-14}$ N/m$^2$.
A: It was during the period called the dark ages. Between 3000K and 60K the Stars had not formed, thus dark. Times between 380000 years to about 150 million years after the Big Bang. See the chronology at https://en.m.wikipedia.org/wiki/Chronology_of_the_universe
3000K was roughly recombination, and then atoms started forming. For most of that time it was mainly hydrogen and the little radiation around was from hydrogen emission. 
If you want to calculate the equations are not that complex. First the relation of temperature T to redshift z is T= 2.72 (1+z), where redshift now is 0. Going back in time redshift will increase. At z = 1100 you get the recombination, approximately, with the T about 3000K. You can calculate for 60K, z about 20. It starts getting cold. And dark. For 300K is looks like z about 100.
For the times for each you can use my answer at Why is it said that photon-wavelengths have increased by a factor of 1000 since our universe became transparent to light? for the lookback times. It also has a time vs z relation.
The derivations are in eg Dodelson and others. I'll have to look up tomorrow the pressure equations, sorry. At 300K it should have some radiation, maybe not too much. At that temperature most atoms are stable, and not that hot to radiate. 
