The short answer is no. The alignment occurs, but the eclipse does not, due to the distances and size differences involved.
There are a few things to consider here. You already mentioned a lot of them, I'll address those considerations separately.
are alignements of the Sun and two planets actually possible (and if so, how frequent)?
We can intuitively gauge this, as we've seen this happen during the transit of Venus:
This situation describes your scenario. The sun, Venus and the Earth are aligned (within some reasonable range).
But, in fairness, you're asking about any of the planets, not just Earth and Venus. So is this possible for any other (or possible even all) combinations of two planets?
This is two questions in one: Do two planets' orbits overlap, and do these planets ever find themselves at this overlap point at the same time?
Do two planets' orbits overlap?
Quite interestingly, not only can they overlap, they must do so. The only way for two orbits to never overlap would be if they were parallel to each other (e.g. one orbit around the equator and one orbit around the tropic of cancer), but that second orbit is impossible.
All orbital planes must intersect with the parent body's center of mass. It's not possible to have an orbit that does not. To put it into visual terms:
There is no way to position these two orbits in a way that they (a) are physically valid and (b) not have any overlap.
Do these planets ever find themselves at this overlap point at the same time?
The only way for them to not find themselves at this overlap point would be if their orbits were periodic. I would argue that perfect periodicity is unlikely to occur in nature, but that's an intuitive argument.
Looking at the planets' orbital periods, none stand out as being particularly periodic. I'm unsure if we can ever prove that our measurements or orbital period at any given time have perfect precision to conclude that perfect periodicity exists.
Other answers have already indicated the frequency with which such overlaps could occur. I'm mostly interested in whether they can occur or not, to see if we can conclude that your idea is possible or not.
Casting a shadow
So far, we haven't actually disproven the theory. Orbital planes will overlap, and our planetal orbits are not periodic so they will eventually overlap. But now we get to the other part: the shadow. Will the shadow of the nearest planest reach the furthest planet?
Let's look at the expected size of the shadow. Do you think it will be larger or smaller than the planet who casts it?
The simple rule of thumb here is that if a shadow is larger than the object itself, then the object itself must be larger than the light source. If instead the light source is significantly bigger, then the object's shadow will shrink, and effectively disappear past its focal point.
Note: I call it a "focal point" because of what it looks like on a diagram, and for lack of a better name. If anyone has a more official term, I'd be happy to hear it.
To put it in visual terms:
I don't think we need to explain how much bigger the Sun is compared to any of the planets.
The relative size of the Sun vs any planet renders the penumbra negligible. It could possible be perceived by lightsensitive equipment (edit: I don't know if it can be perceived, I just can't prove that it can't be perceived), but not by a human who observes this while standing on the further planet. It won't look like an eclipse. I refer back to the transit of Venus:
Given the significant size difference between the Sun and planet, and the massive distances between the planets themselves, it doesn't seem feasible for the umbra to even reach the next planet's orbit before it hits its focal point, which is why we don't describe such a transit as an eclipse.
NASA explains this on their website:
Like an eclipse, a transit occurs when one object appears to pass in front of another object. But in a transit, the apparent size of the first object is not large enough to cast the second into complete shadow. Instead a much smaller dark shadow makes its way across the face of the further planet or star. Perhaps the most famous recent transit was that of Venus across the face of the sun in 2012.
Casting a shadow - intuitive explanation
There's a more intuitive way to express this. If a certain object (nearest planet) can block a light source (Sun), that means that from the point of view of the observer (on the furthest planet), the object must appear to be bigger than the light source. There's a quite well-known scene from Apollo 13 that showcases this:
Tom Hanks (as Jim Lovell) is blocking out the moon using his thumb. Even though the moon is much bigger than Tom Hanks' thumb (citation needed), the relative closeness of his thumb to his eye (compared to that of the moon) makes the thumb appear bigger than the moon.
Let's say that Tom Hanks puts down his hand, and Chris Hadfield sticks his thumb out of the window of the ISS. His thumb is perfectly aligned between the moon and Tom Hanks' eye. We assume his thumb is the same size as Tom Hanks' (citation needed).
This wouldn't block out the moon anymore from Tom. But why is that?
Very simply put, even though the thumb is the same size, the relative distances between the eye, thumb and moon have changed. And now, Chris Hadfield's thumb does not appear to be bigger than the moon, when observed by Tom Hanks. And therefore, it can no longer block the moon from Tom Hanks' view.
So, we can reformulate your eclipse question: Is there any planet which appears to be bigger than the Sun, when observed from another planet?
The answer is no.
Edit: I've decided to make the below text easier to parse, by talking about the Earth and Venus. These are just examples, the same applies to any combination of planets, where Earth = furthest and Venus = nearest.
If you assume that the Sun is
X times larger (in diameter!) than Venus, and the sun is
Y times further from Earth than Venus is from Earth, then Venus will appear to be bigger than the sun (when standing on Earth) when
X < Y.
In other words, when measuring the distance from Earth, the sun needs to be further away than Venus, by a larger factor than the sun is bigger than Venus.
As a simple example, if the sun were exactly 5 times bigger than Venus, it would have to be >5 times as far from Earth as Venus is, for it to appear to be smaller than Venus.
If you look up the numbers for planet diameters and orbital radii, you'll notice that this doesn't happen in our solar system. Not even close. This is because the sun is simply too big relative to any ratio of two planetary orbits.