Great question! To start off, you’re exactly right that an “equals sign” in a physics equation has nothing to do with causation. The equations just tell us about correlations in both space and time. Correlation is not causation.
The realization that you can’t find causation in the ``bare correlations’’ has been a key piece of the huge increase in our understanding of causation over the past 30 years. So you can’t find causation in physics equations. Instead, along with the correlations, you need something called a causal model. Everyone uses causal models in physics, but they’re rarely explicitly represented. Physicists tend to just write down the equations and claim that this is all there is, but of course that’s not true: we use the equations in different ways and different contexts. Using the equations in some particular way is what we mean by a “model”, and physics models are almost always causal models. There absolutely is causation in physics models, and it deserves to be more widely understood. It’s not just random whims, and it’s not time-ordering (we have another set of words for describing time-order). Instead, it has to do with real or modeled interventions on a system.
When physicists have a working model of a finite system in spacetime, we intuitively make it a causal model. Specifically, we take some particular physical situation and then imagine how it might be made different. And we imagine the difference-making as originating from outside the system – some parameter on which it seems natural to “reach in” and “intervene” upon, changing it to some new value. This altered parameter is naturally thought of as a “cause”. All the other parts of the system that are correlated with this intervention is the “effect”. This is the interventionist account of causation in a nutshell: in the lab, the causes include our experimental interventions. More generally, causes are the variables one imagines directly intervening on, from outside the system. (Technical caveat referenced below: when we imagine intervening on a variable we also perform “arrow breaking”, preventing other causes of those same variables; we can’t always do this in the lab.)
Consider your example about how the electric field $E$ is correlated with $Q$ and $r$ and $k$. It’s probably intuitive that the position of the charge causes the E-field, and that this is true even if there are other E-fields around. That’s because it’s easy to imagine reaching in and intervening on the precise position of the charge, and this difference would be correlated with the corresponding E-field everywhere. So the charge position causes the E-field. Conversely, imagine reaching in and increasing the E-field in some region of space. The model you’re describing wouldn’t see an (instantaneous) change in the position of that charge, so one would not say that the E-field causes the charge position. (You could add a few more equations where an external E-field would cause a force on the particle, but then you’d have to distinguish between the various E’s in your various equations; each of them could have a different causal assessment.)
Interestingly, some people could look at your equation and will not see $Q$ as causing $E$. Other people might. That’s because people will disagree on whether it’s reasonable to imagine reaching into the model and “intervening” on the value of $Q$ directly. Some simulations let you do this, at which point it seems clear that $Q$ causes $E$. But if the particle is an electron, you might not see $Q$ as being “changeable”, even in principle, so $Q$ starts to be treated more like “k”. It's also barely possible to imagine “$k$” causing $E$, but such a model would be very unusual; most people won’t think of $k$ as a cause of $E$, since it’s hard to imagine some external way to change it.
On your point about relativity: don’t confuse the map for the territory. Relativity tells us where we might expect to see causal relations, and where we definitely do not expect to see causal relations. But it doesn’t say what causation is in the first place. Knowing where we might see causal relations is a far cry from a definition of causality. In fact, if you treat this as a definition of causality (as far too many people tend to do), it becomes circular; the restriction on direct causation outside the lightcone becomes meaningless, if you take that restriction to be the very definition of causation. Only by supplementing relativity with the above definition of interventionist causation can you actually draw some real conclusions.
Entropy considerations come into a causal model in two different ways. First, it comes in via the external agents in a causal scenario, real or imagined. Real external agents can intervene on a system, “causing” something in that system, and entropy considerations put constraints on what those agents can actually do. They can directly set the input of a device, but they can’t set the corresponding output. For imagined agents, as in most physics models, it gets a little trickier, but our understanding of entropy usually constrains our imagined interventions on a system. We can easily imagine changing the initial state of a system from outside the system, but it’s much less common to imagine imposing an external future boundary condition. If you did, you would find that the causal effect was opposite the order we usually expect (although note that the direct causes, even then, would still lie inside lightcones).
Second, entropy comes in when we “break arrows”, in the causal modelling lingo. When we imagine intervening on a parameter, it’s crucial that nothing else is also thought to cause that parameter, or it messes up our causal assessments. When we set an initial condition on some variable in a system, we don’t allow other things in the past to thwart our choice for that variable. So in our models we "break the arrows" linking the variable to events in the past, but not to events in the future.
Take the charge example. At $t=0$, I want to relocate it from $x=0$ to $x=1$ to see what will happen in the model. But I have equations telling me that its past position at $t=-1$ and the forces acting on the particle between $t=-1$ and $t=0$ actually determine its true location at $t=0$. But for causal purposes, I want to “break” these equations; ignore them, set them aside, and instead imagine I have complete freedom to choose $x=1$ at $t=0$. But we don’t generally do the same for the future; we don’t break the link between forces and motion after $t=0$, allowing the model to follow those equations. This seems perfectly reasonable to us entropy-increasing agents, but of course it breaks an arrow of time. If I instead broke the future equations, and left the past equations in place, I’d find that my intervention at $t=0$ was actually a final boundary condition on the $t<0$ side of the model, and I’d see entropy-decreasing behavior. Such a model doesn’t match reality, so it’s an empirically bad model, with an objectively poor choice of imagined interventions.
In short, entropy tells us which causal interventions to consider, and which causal arrows to break. If we pick the wrong interventions and the wrong arrow-breaking, we’ll get results that contradict our entropy-increasing observations, telling us that we probably shouldn’t consider those interventions in the first place. But the same is true for physics equations; if we pick the wrong equations, nature tells us that we’ve made a mistake. Using the scientific method to compare causal physics models with reality therefore guides us towards better equations, as well as guiding us towards the actual causes of a given system. We do this automatically, so that we hardly think about it; we just “know” that we should use initial conditions in our models, a lesson that we’ve already long since internalized. But just because this choice of initial-boundaries is now coming from us when we use our physics models doesn’t mean that it’s purely subjective. It’s the objective considerations described above that tell us which interventions are reasonable to consider in the first place. So in this sense, the direction of causation is just as objective as our most successful equations; they can both be determined empirically.
