Can you model cold as flowing? Obviously, cold isn't a "Thing".
Coldness is the absence of heat, and when you add a cold ice-cube to a drink there are no particles of "coldness" flowing into the rest of the drink cooling it down - the heat of the rest of the drink (in the form of molecular kinetic energy) is flowing into the ice-cube, leaving less left in the resulting drink.
I understand all of that :)
But my question is:
If you were to model coldness conduction as a thing that flowed (just as we model heat conduction as a thing that flows), would you get accurate macro-level models?
Can you accurately predict, say, the effects over time of placing an ice-cube on the centre of a thin aluminium tray, by modelling the cold as flowing out from the the ice-cube. (In the same way that you could predict the effect over time of placing a cube of heated metal in the same place)
If yes, are the equations and/or constants different, or is it just "add a minus sign"?
 A: The way I've personally seen heat transfer approached, there is nothing wrong with thinking of it that way.  In some sense, you arent even thinking of it any differently than it is typically taught.
Heat transfer and "cold transfer" are fundamentally identical; they describe the same process.  Heat is energy in transfer and therefore always involves flow of heat from one system to another.  Anywhere with heat flow has an equal and opposite "cold" flow.
Really, it is about reaching a thermal equilibrium.  There is nothing more privileged about the "positive" side of the energy flow compared to the "negative" side.  It is just an exchange of energy, and we usually formalize things from the side where more energy is a greater value than less.
A: The main difficulty in describing heat as a flow is that, unlike mass or charge, heat does not satisfy the continuity equation,
$$
\frac{d\rho}{dt} + \nabla\cdot\mathbf{j}=0,
$$
where $\rho$ is the spatial density of heat/charge/mass, whereas $\mathbf{j}$ is the heat flux/electric current/hydrodynamic flux. The reason for that is that the amount of heat is not conserved, it can be generated and disappear at any point in space. The equation of continuity thus needs to be modified, to include heat sources and sinks, but this makes it too general to be useful (at least most of the time).
Still, analogy with the flow is useful, especially when one wishes to separate the processes of heat generation and its displacement in space, which is why we often speak of heat flow, heat flux, heat transfer.
The quantity that is indeed conserved is the energy, which consists of heat and work. In this sense the conservation of energy is one of the conservation laws that constitute the basic equations of fluid dynamics.
A: 
If you were to model coldness conduction as a thing that flowed? ... If yes, are the equations and/or constants different, or is it just "add a minus sign"?

"Add a minus sign" is ambiguous. First, you'd need to define a negative energy that flows in the opposite direction to energy, and this would affect all heat flow relations.
Notably, heat transfer also transfers entropy—that is, entropy is the thermodynamic conjugate variable to temperature, so temperature differences drive entropy flow—so you'd also need to define a negative entropy that flows in the opposite direction as entropy. (This breaks Boltzmann's formula, or perhaps forces a revised Boltzmann's constant that's negative.)
Then, you'd need to account for the fact that spontaneous processes generate entropy in accordance with the Second Law, so the negative entropy would need to incorporate this behavior in terms of a sink.
Furthermore, anything that dissipated heat would need to be modeled as a sink for both the negative energy and the negative entropy you've defined.
This all seems more likely to produce great confusion than great clarity.
Edit: There are some questions seeking clarity about the aspects of entropy transfer and generation.
Entropy is both transferred and generated during real heat transfer. (In reversible heat transfer, it is only transferred, not generated.) The transfer part is directional according to which object is heating which other object. The generation part is symmetric and independent of what’s heating what; it depends on the flux magnitude, not the sign. Changing the framework from heat transfer to “cold transfer” needs to incorporate all of this behavior.
Put another way, any real heat transfer generates entropy $S_\text{gen}$ (as well as transferring entropy $\Delta S$ and energy $\Delta U$). $S_\text{gen}$ and $\Delta S$ have the same sign only for the object being heated. If you flip the signs so that $U^\prime=-U$ and $S^\prime=-S$, then positive $\Delta U^\prime$ switches to the object doing the heating, but the signs of $S^\prime_\text{gen}$ and $\Delta S^\prime$ are still the same only for the object being heated. This breaks the symmetry of a simple sign change.
A: I really think you messed up lots of things here. Moreover, your qualitatively description of "your interpretation of Physics" doesn't help. I start with some clarifications about the confusion about Physics or poor language for describing it, and then explain a way to build a model for "cooling transfer" and "cooling source".
Clarifications

*

*temperature vs. heat: temperature is what you measure with a thermometer and it is the macroscopic thermodynamic variable related to the kinetic energy of molecules; instead, heat is not a thermodynamic variable, it is a way to transfer energy, and it depends on the interactions between molecules;


*hot/cold describes temperature: so hot/cold are qualitative adjective that refers to temperature, not to heat or heat transfer. When I read

Coldness is the absence of heat

my eyes started bleeding. Absence of heat transfer means absence of a mechanism that can make the energy change. Put hot coffee in a thermos (vacuum flask): approximate there is no heat transfer, and your coffee will remain hot. This is just an example of no heat transfer, hot system;


*heat doesn't flow, heat is a mechanism to transfer energy: there are several heat transfer mechanism of different nature, mainly conduction (diffusive nature, in PDEs usually model with a Laplacian term), convection (in fluids, due to mass transfer when a fluid moves, usually modelled with a material derivative), radiation (typically relevant for high temeprature, being proportional to $T^4$);


*heat sign convention, and the "cooling" transfer mechanism": the convention of sign says that a heat transfer to a system is positive if it makes the energy of the system increase, and with this convention you can write the First Principles of Thermodynamics as $\Delta E^{tot} = W + Q$, while the Second Principles of Thermodynamics (in the Clasius version) reads $d S \ge \delta Q /T$. One of the implications of the Second Principle is that if you take two bodies at different temperature $T_1 > T_2$, the spontaneous heat transfer observed in nature makes the energy of system 1 decrease and the energy of system 2 increase;
Cooling teansfer and cooling source model
Once everything above is clear, we can move on to the definition of "cooling" transfer and cooling source.

*

*if you really want to introduce the definition of a "cooling transfer": you can define it adding a minus sign before the conventional heat transfer, $C:=-Q$. Fortunately, Physics doesn't depend upon your definitions, so First and Second Principles respectively read $\Delta E^{tot} = W - C$ and $d S \ge -\delta C /T$, and other physical laws transform in the same way: as an example, Fourier laws for conduction reads $\mathbf{q} = - k \nabla T$ (heat flow) or $\mathbf{c} = k \nabla T$. In both cases, when Fourier's law is used in energy balance equation, you always get
$\partial_t T - \nabla \cdot ( k \nabla T ) = r = -c$, being $r$ the heat source volume density, having defined the "cooling" source volume as $c=-r$.

