What would happen to a lone quark? If a quark simply popped into existence, what would happen to it? Or is this question unanswerable for some reason (e.g "a quark, by definition, exists with other quarks, and has no mathematical form on its own")?
I take lone to be on it's own in space, without interaction with the environment
 A: We are talking standard model of particle physics and quantum mechanics. The basic question is "can there be a lone quark"?
Quarks are components of hadrons. Lets see the analogous, "can there be a lone electron" ?
Yes if enough energy is supplied. When in an aggregate of atoms, one can have the plasma phase . This is easily achieved because it is the electromagnetic interaction where the coupling constant  is small, and there can be solutions with the electrons free of the atom .

Quarks are mainly bound in the hadrons with the QCD interaction, where the coupling constant is 1, and if one tries to free a quark an enormous amount of energy is needed.
Nevertheless, in the cosmological models, a quark-gluon plasma is hypothesized when the energies were very high.

Quark–gluon plasma or QGP is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word plasma signals that free color charges are allowed. In a 1987 summary, Léon van Hove pointed out the equivalence of the three terms: quark gluon plasma, quark matter and a new state of matter. Since the temperature is above the Hagedorn temperature—and thus above the scale of light u,d-quark mass—the pressure exhibits the relativistic Stefan-Boltzmann format governed by fourth power of temperature and many practically mass free quark and gluon constituents.

In this plasma the quark may be considered "lone", continually exchanging gluons within the plasma.

I take lone to be on it's own in space, without interaction with the environment

As far as the present physics model goes, the answer is no. Due to the coupling constant being 1, it is not possible to have a free quark the way you have a free electron. In order to be free, energy must be supplied, and the further away from the other quarks, the more energy has to be supplied, which allows to have the quark gluon plasma at very high energies.
A: A quark can't simply pop into existence. Color confinement applies to all processes, including processes that create and destroy particles. Quarks are only created and destroyed in color-neutral combinations.
If you altered the physical laws so that a lone quark could pop into existence, then it would not be color-confined, but that isn't very interesting because you explicitly repealed the law that forces confinement.
People sometimes say that quarks are free at high temperature. They are free in some sense, but they aren't free in vacuum. There is a bag model of protons and other hadrons where the quarks move semi-freely inside the hadron, but can't leave it. A quark-gluon plasma is a larger bag containing more quarks. The quarks move semi-freely in the bag, and in that sense are "unconfined", but they still can't leave the bag, and the bag still has to be color neutral.
In the early universe there was a quark-gluon plasma that filled all of space. If it had an edge, it was so far away that we can't see it. All quarks were quasi-free in that epoch. People sometimes say that as the universe cooled, the quarks joined together into hadrons, but that's wrong. What happened is that the universe-sized hadron split into smaller pieces. The pieces contained a minimum of two quarks.
