Or does the strong force go back to zero at distances say larger than an atom?
The charge of the strong force is called color, and outside particles like protons the total color charge seen from the outside is zero. It stays zero because of color confinement - at some point, it's less energetic to create color-anticolor pair instead of allowing color imbalance.
Of course there are also residual forces from a color-neutral bound state to quarks which are a bit further away, but those are not nearly as strong. They are for instance responsible for holding nuclei together.
The important property of the strong force here is that it interacts with itself. The field of two quarks together is not the sum of their individual fields, because the force-carrying gluons are also color charged and thus interact with each other.
My understanding is that if you do manage to pull two quarks apart the energy of separation is eventually enough to form two new quarks, thus pairing them up again.
My question is whether this pair production happens before or after the strong force reaches the region where it is constant?
I think (guess?) that the strong force does the same, in the sense that there is a "neutralizing tendency", anyway, but the mechanism is different. Protons and neutrons, and pions too, are neutral when you watch them from far away. (But not so, when you are trying to explain why atomic nuclei are formed, protons and neutrons _like_ to clump together.)
The difference between a photon field and a gluon field is that gluons are attracted to each others in a way that I, as a non-nuclear-physicist don't quite understand (as the symmetry between them is not something I've actually studied); but as photons form "beams" (you can think of them in a linear algebra sense of vectors, almost!) that propagate in a very geometrically uniform way, gluons, being attracted to each others, form "tubes", which behave, as any self-interacting system would, in a very dynamic way.
Imagine a cellular simulation (like the Game of Life, but more... floating point instead of of integers & squares) where each quark sends gluons but they tend to clump together and form tubes? These tubes can't be super long, because quantum physics and the universe works in a way that energy gets minimized, and anything that could happen to make that happen, tends to happen. That means that if there's enough energy stored in the tube, it becomes "cheaper" for the universe to sever that connection and instead, produce new, separate particles that have their own, internal "tubes". That means that no long "tubes" of gluons are allowed in the universe, and thus, the strong force of the Strong Force is contained.
So the mechanism seems to be really super different from "overlapping +/- fields", in a sense, but the result is the same: no forces (albeit small residues) seen from afar.