Atomic gravitational field? This might be a silly question, but if every atom has its own gravitational force could atoms or molecules be attracted to each other over vast distances in the void of space if there were no other greater forces being applied? Is this similar to the cooling/slowing theory?
 A: Yes indeed; all atoms are attracted by all other particles in the universe by gravity. However, as gravity depends on mass (or energy content, rather), these forces will usually not be very big. 
For example, if you have two protons (identical to ionized hydrogen atoms), placed 1 meter apart at rest with respect to each other, the gravitational force they exert on one another is 
$$
\begin{align}
F_{grav} &= G \frac{m_1m_2}{r^2}\\
&\approx \left( 6.67384\cdot10^{-11}\,\text{m}^2\,\text{kg}^{-1}\,\text{s}^{-2}\right)\cdot \frac{\left(1.67262178\cdot10^{-27}\,\text{kg}\right)^2}{(1\,\text{m})^2}\\
&\approx 1.867115 \cdot 10^{-64}\,\text{N}
\end{align}
$$
while the repulsive force due to their electrical charge will equal 
$$
\begin{align}
F_{electrostatic} &= h_0 \frac{q_1q_2}{r^2}\\
&\approx\left(8.9875517873681764\cdot10^9\,\text{N}\,\text{m}^2\,\text{C}^{-2}\right)\cdot\frac{\left(1.602176565\cdot10^{−19}\,\text{C}\right)^2}{(1\,\text{m})^2}\\
&\approx 2.307077\cdot10^{-28}\,\text{N}
\end{align}
$$
In other words, the electrostatic force between them is 36 orders of magnitude stronger than the gravitational force! To understand the universe, the much more relevant question is therefore whether all positive charges in the universe are balanced against all negative charges, in other words, whether the universe is electrically neutral. Most cosmological models assume that it is (an assumption that turns out to make many models agree with observation), however, AFAIK, this is still an assumption without any hard, direct evidence. 
Even if you would use non-ionized tritium atoms, their gravitational force would still not be stronger than their electrostatic force; the negatively charged electrons of the two tritium atoms are slightly closer to each other than their nuclei (or rather, have a higher probability to be). Repeating the calculation with 2 neutral tritium atoms, taking into account the Bohr radius, the electron mass and 2 neutron masses per atom, 
$$\begin{align}
F_{gravity} &\approx +1.68101\cdot10^{-63}\,\text{N (attractive)}\\
F_{electrostatic} &\approx -4.48416\cdot10^{-44}\,\text{N (repsulsive)}
\end{align}
$$
This second force will however fall off with distance faster than gravity, so that at large distance, gravity is indeed the only force remaining. 
This is a simplistic but good enough representation of the magnitudes of the forces involved, and I hope that you are convinced that gravity is really a minute factor in most interactions, and usually completely safe to ignore.
However, it is still a very important force, but for other reasons. To answer your second question: if the entire universe were made up of normal hydrogen gas  (not ionized, so electrically neutral), their mutual gravity by itself would in principle not slow them down (cool them). Since gravity is a conservative force, the total kinetic energy in the universe would stay the same, regardless of any gravitational interaction between the atoms. This would only change when two atoms collide, which, given the average mean-free-path of particles in the void of space, is a pretty rare event. Therefore, the rate at which the thermal energy (= average kinetic energy) would be converted in to electromagnetic radiation (= what kinetic energy would be converted into; assuming low average impact energy) would be pretty slow. 
However, their gravitational interaction would cause the gas cloud to become non-uniform; regions of slightly higher density ("lumps") will start to appear. These lumps will become denser, because there is more mass in the same place, and thus stronger gravity. This will attract more and more hydrogen atoms towards the center, and making collisions far more likely. Given a few dozen million years, a star will be born. 
And given 13.77 billion years, human beings will start to appear, pondering about how it all came about :) 
