From what I understand, the energy or temperature of a structure with mass has no effect on the gravitational force it emits. Is the same true for the other fundamental forces?
What would happen to the strength of nuclear force and electromagnetic force of an atom if I lowered it's temperature to infinitesimally close to absolute 0? Or even absolute 0? Is there a direct correlation between temperature and strength of the forces.
Temperature is a variable defined over an ensemble of particles. The forces of the standard model describe the interactions between individual particles in a mathematical manner. These interactions themselves cannot be changed as such.
What can change is the outcome of the interactions because they depend on energy and momentum. In a particle ensemble higher temperatures are connected with higher kinetic energies of the particles it is composed, so for very high temperatures the interaction effects will be different than for very low energies. For example, the cosmological model accepts that when the temperatures are very high a there are no nucleons but a quark gluon plasma appears.
For temperatures close to absolute zero there is no energy for individual particles to interact.
In principle, everything in the Universe affects everything else. On the other hand, the strength of these "nonlinear" effects is typically exponentially suppressed. In order to become noticeable, one should overcome some characteristic threshold. So in the case of electromagnetic interactions, you can imagine that if the temperature $T$ of the system you atoms is in (think of some cavity filled with radiation) is such that $k_B T> 2 m_e c^2$, where $m_e$ is the mass of an electron (the lightest known charged particle), then your vacuum will be full of fluctuating electron-positron pairs, which will very efficiently screen any charge, thus turning your long-range Coulomb interaction into a short-ranged screened one.
The same goes for all other interactions, provided the temperatures reach the levels $k_B T \sim Mc^2$, where $M$ now is the mass of the lightest particle that feels the interaction in question (has non-zero corresponding charge).
