Multistage coilguns make no sense

Other stages could probably steal some magnetic flux from the current firing stage, even creating magnetic field in opposite direction, althought they probably are not closed cicrcuits when current stage fires, so inducing current inside of them shouldn't be possible. Also, if this was the case, the effect would be different for different materials,stage lenghts, while it is the same.

I also observed something interesting: ingame peak acceleration is almost exactly inversely proportional to stage count. In case of problems with magnetic circuit, the behaviour would be surely much more complex and dependent on many parameters. 

You are correct in that they do not work in the manner that would be scientifically accurate, but efficiency is not a function of distance, it is a function of time, if the projectile gains the most velocity in the first stage then the second stage while having a larger distance to work on will have less time and since it shows Gs which are ~9.8m/s^2 the shorter time will result in less m/s.

Also open circuits do conduct power, often not much, but enough to create havoc, this is especially relevant in the vicinity of nearby fluctuating magnetic fields, but I don't think the game models that (or if it does it does it in a simplified manner since the finite element method solutions for these kind of problems would probably not be leaving much room for computations of the actual game).

Yes, it is a capacitor one, and it is obviously not tuned correctly, but in this case it does not matter at all. I use it for demonstration purpose, not as a practical weapon. The only important fact is that second stage is more efficient than first, and the game still says that this coilgun is more efficient when you remove it. 

Acceleration can be function of anything, because acceleration is a vector representing how velocity changes over time, but it does not need to be dependent on any of those. There is no reason why acceleration shouldn't be function of result of dice roll done by captain of the ship, or function of total elephants count. But in this case we have a PLOT representing how acceleration changes over distance. This means that there is function, which for every distance belonging to a (0, barrel length) returns some acceleration, so acceleration is in fact a function of distance (I do not say that it is not function of anything else). Given that there is that function, and knowing that projectile moved in a single direction, we can now calculate work as W = integral(acceleration(distance) deltadistance) * mass, which is the blue area under the plot multiplied by mass of the projectile.

First of all: the science!
Work is defined as a function of distance. Distance itself can be expressed as a function of velocity or acceleration. The importance of this nuance is that work is always done over distance. If you characterize the distance traveled as a function of time, the start and end points of your integral themselves become functions of time.
As for efficiency, if a second stage that is identical to the first is used to further accelerate the projectile, it will be less efficient, because the projectile enters with a higher velocity, and thus has less effective distance (compared to starting from a standstill) to accelerate. However this can be counteracted by spacing the coils or modulating pulse length. It is important to realize that the work put into the projectile by multiple identical stages is always the same. Changing pulse length or spacing coils means adding more work to the projectile, as was pointed out already!

Second: implementation in CoaDE.
The way that capacitors are implemented currently, especially for coil guns with multiple stages, is messed up. This is easily demonstrated by makin capacitor coilguns with >100% efficiency.
Multiple stage coilguns are broken because the force applied is inversely proportional to the number of stages. Personally, I think this was done, because stage efficiency is additive, at least (possibly multiplicative), as far as the work applied is concerned. Efficiency of multi stage coilguns with >100% per stage efficiency would thus quickly escalate.
Another possibility is that the stored energy isn't equally distributed across the stages. It looks like it is, but taking into account acceleration in capacitor railguns, it's obvious the game makes some weird assumptions as far as power input is concerned.
Regardless, what is physically realistic isn't really relevant here, because the efficiency or efficiency loss in this case has more to do with the formulae the game uses, rather than what physics dictates should happen. With out access to these formulas, finding the efficiency sink in multi stage coilguns is more down to luck than judgment.

You calculations aren't wrong. Because both capacitors discharge completely, both stages input the same work, and thus should be equally energy efficient.

As for the game violating physics. This is completely necessary. To properly calculate the exit velocity of a coil or railgun you would need to run a fluid dynamic simulation of charge flow under maxwells equations. This takes days on computers set up to do so and would probably take days on desktops. Thus simplified equations are used. These create artifacts, such as singularities (aka points of infinite parameter compression). There is obviously such an artifact in multi stage coilguns. That the force decreases with the number of stages is either a result of or a way to counteract this artifact.

Edit: as an example: if stage efficiency is multiplicative, and stage efficiency is 130% (which I have achieved) then 2 stages are 169% efficient, 3 stages are 220% efficient and so on. At 100 stages this results in an efficiency of 2.48*10^13%. This is far more detrimental to the simulation than multi stage coilguns being ineffective.