At a high level, distinguishing cause and effect is typically easy enough: the cause comes first. I drop a ball off a roof; therefore, it falls and hits the ground. But on a fundamental level, physics is local; thus, it seems that, as you get more fundamental, the temporal interval between cause and effect approaches zero. We say that forces cause acceleration inversely proportionate to mass, but at that level the action and reaction are simultaneous. Why (if at all) would it be wrong to say that acceleration causes forces proportionate to mass?
I would suggest that one way to distinguish cause from effect when the two are simultaneous is through material implication.
That is, if at some time two events A and B occur, the cause is that one which implies the other. So, if A being true means B must be true, then A is in some sense the cause of B (I realize that the whole correlation versus causation problem comes up here, but I am only trying to provide a rough model).
For example, suppose that at some time t we have a big spherical mass M (event A), and from a distance r away you (mass m) experience a real force equal to GMm/r^2 (event B). In this case, A implies B by Newtonian gravity. However, B does not imply A: there could instead be a number of different masses that happen to produce that force.
For your example specifically, I think it is important to flesh out the definitions. We say that real force causes acceleration inversely proportional to mass. Fictitious force, on the other hand, is caused by acceleration on a frame (which is caused by a real force or another fictitious force). So, if for that example we have a real force on an object and an acceleration in that object simultaneously, the real force implies an acceleration, but the acceleration does not imply a real force: it implies either a real force or a fictitious force. If instead we have a fictitious force on an object and an acceleration on that object's frame simultaneously, the fictitious force does not imply an acceleration on that specific frame (it could be any frame that happens to contain the object), while the acceleration on the frame does imply a fictitious force on the object. This may seem a bit picky, but pickyness can make all the difference when you're talking about simultaneous events.
I think this is the crucial difference between cause and effect: not temporal separation, but material implication. After all, that's why it's called "cause" and "effect", not "before" and "after".
Nothing at all. This is the same base idea at the heart of non-inertial reference frames.
This is like Einstein's claim (at the heart of his idea that grew into relativity) that it is impossible to distinguish between an accelerating frame or reference and an inertial frame of reference in a gravitational field. That is to say that if we are in an elevator that is accelerating upwards at 2 m/s^2, or on a planet that happens to feel 12 m/s^2 of gravity (that is to say that the force or the acceleration are indistinguishable).
The finite speed of light puts the whole idea of simultaneity into question. For example consider a person A standing in a field halfway between two poles (A+B). He sees a lightning bolt hit each pole simultaneously. He thinks it's simultaneous.
Now person B is standing closer to one of the poles (B) than the other. What does he see? The lightning strikes are not simultaneous at all. Pole B is hit before pole A.
There are many other such thought experiments, but the final conclusion is this:
Fact: simultaneity is relative to the observer because of the structure of our universe. It is not an absolute that two events observed by Mr A to be simultaneous will be observed by everyone as simultaneous. There is not such thing as an absolute clock which can be used to measure these things or determine simultaneity.
Now consider causality. Einstein showed that the order of cause and effect is preserved no matter where the observer is because causes cannot propagate faster than the speed of light. This preservation of the order of cause and effect is again necessary in physics. If it were not so we could have effects occurring before causes. That is obviously nonsense. If you want more on this, look up the Minkowski Diagram.
Fact: Cause and effect order are not relative to an observer. The order holds in all frames of reference.
Therefore: It is a fundamental that simultaneous cause and effect is impossible. The relative nature of simultaneity conflicts with the fundamental requirement that effect cannot precede cause.
Force is an auxiliary concept in Newtonian mechanics, and if you like, you can simply dispense with it. For example, rather than saying that F=GMm/r^2 acts on m, and m's resulting acceleration is a=F/m, you can eliminate F algebraically and say that a=GM/r^2.
The usual way of thinking about causality in physics these days is that you want existence and uniqueness of solutions to Cauchy problems. In other words, we don't think of forces causing accelerations, charges causing electric fields, and so on; we simply think of initial conditions causing final conditions. Re this notion of causality, you may be interested in this paper: John D. Norton, "Causation as Folk Science," 2003. It seems to have been widely discussed, but not necessarily widely accepted. You can find a lot of references by googling "norton's dome." There are good papers by Korolev and Laraudogoitia. This paper is also interesting: Z. Xia, “The Existence of Noncollision Singularities in Newtonian Systems,” Annals Math. 135, 411-468, 1992. There is a good survey in Earman, A primer on determinism, 1986.
I think the biggest difference between cause and effect is that it is the cause that is taking the initiative, not the effect.
In other words, the cause can decide whether the whole cause-effect thing happen or not. Take the acceleration example, the reason you can not take the accelaration as the cause is that the acceleration can not happen on its own will - it has to depend on the force to happen, but the force can happen on its own, whether or not to cause the acceleration.
You cannot have cause and effect occurring in the same place (down to less than the diameter of a proton) and at the same time. Cause and effect (the two lightnings) would be the same photon. Cause and effect would clearly be indistinguishable.
The lightning analogy is valid.
Simultaneous cause and effect is impossible in a relativistic universe.
Locality in physics is an observation that we don't have 'spooky action at a distance' as Einstein so eloquently stated. In QM we do have such spookiness, which opens up a whole 'nuther can of worms when you are talking about causality. It's also the basis for a scientific attack on the Kalam Cosmological Argument.
The whole thing is neatly wrapped up by Bell's Theorem which shows the conflict between relativity and QM.
For example of the problems with asserting laws of causality, an experiment conducted in the future can influence results you get in the present.
This is the basis for saying there are uncaused events in QM; the arrow of time is not in question, and all reasonable notions of causality require events occur after causes in the temporal sense. Relativity imposes the additional constraint that the time between the cause and event exceeds the distance / speed of light.
Radioactivity is the most often cited example of acausal events in physics. It caused Einstein much angst. QM however, aided by Bell's theorem seems to be winning the argument.