Can a planet in our system eclipse the sun as seen from another one? When the Sun, Earth and Moon get properly aligned, we get eclipses, where the Sun is partially or totally hidden by the Moon when seen from Earth.
Is it possible for one planet of the solar system to produce an eclipse on another one?
There are two underlying questions in this:

are alignments of the Sun and two planets actually possible (and if so, how frequent)? Given the slightly different orbital planes of the planets, this seems difficult?

even though planets are much smaller than the Sun, with appropriate distances between the three bodies considered, is it actually possible for one planet to partially or completely hide the Sun like we see in the Sun-Moon-Earth case, or does it only happen for planet-satellite combinations given the relative distances (i.e. in all other cases if would just be a tiny point transiting across the Sun)?


I understand that the Earth would not be able to project a shadow far enough to get such a phenomenon on other planets, but are there other combinations where it’s possible/did happen?
In our solar system, it is possible for one planet to partially eclipse the sun, but it is not possible for any planet to cause a full solar eclipse as seen from another planet.
The sun is too big and the planets are too small and too far apart.  Transits occur, and can occur for any pair of planets, but they are very rare.  As seen from earth, Mercury only transits a dozen or so times per century, and before 2004, the last time Venus had transited the sun was 1882. http://www.eclipsewise.com/transit/transit.html Orbital periods increase for the planets further away from the sun, so transits across the sun by the outer planets (while possible since no planets are in orbital resonance since Pluto got kicked out) are increasingly rare. Michael Seifert dug up these treasures, documenting transits of the outer planets: Jupiter, Saturn, Uranus.  As expected, the transit of Uranus from Neptune is the rarest, occurring next in October, 38172 CE.
Proof that no planets could fully eclipse the sun from another planet:
To rule out the possibility that any eclipses are possible, We only have to verify orbits pairwise in order of increasing distance from the sun. Since, if the $nth$ planet can't fully eclipse the sun from the $n+1_{th}$ planet than it certainly won't be able to do so from any other planet.  For example, if Jupiter can't eclipse the sun from Saturn, it certainly won't be able to from Neptune.
As seen from a planet, if the apparent size of the sun in the sky is smaller than the apparent size of the next inner planet, then a full eclipse is possible.  The largest an inner planet can appear from an outer planet is when the inner planet is at aphelion and the outer planet is at perihelion and the pass is directly overhead of the observer.
The formula for angular radius of a sphere is $\delta = \arcsin (d/2D)$ where d is the diameter of the sphere and D is the distance between the observer and the center of the sphere.
I ran these calculations using matlab and data from NASA. https://nssdc.gsfc.nasa.gov/planetary/factsheet/
For my input data, the rows are planets, and the columns are diameter (km), perihelion (e6km), and aphelion (e6km).
Here are my results.  Since the values in the second column are smaller than the first for each row, no inter-planetary solar eclipse is possible. As an example of how to read this table, an observer on Venus will see an angular radius of 6.48e-3 radians for the sun in the sky, but an angular radius of only 6.47e-5 radians for the next inner planet, which is Mercury.

Here is my matlab code for posterity:
sunDiameter = 1.3927e6;
planetVals = [4879,12104,12756,6792,142984,120536,51118,49528;...
46.0,107.5,147.1,206.6,740.5,1352.6,2741.3,4444.5;...
69.8,108.9,152.1,249.2,816.6,1514.5,3003.6,4545.7];
results = zeros(2,7);
for i=1:7
  innerPlanetDiam = planetVals(1,i);
  innerPlanetApogee = planetVals(3,i)*1e6;
  outerPlanetPerigee = planetVals(2,i+1)*1e6;
  outerPlanetDiam = planetVals(1,i+1);
  sunApparentSize =
    asin(sunDiameter/(2*(outerPlanetPerigee-outerPlanetDiam/2)));
  innerPlanetApparentSize =
    asin(innerPlanetDiam/(2*(outerPlanetPerigee-outerPlanetDiam/2-innerPlanetApogee)));
  results(1,i) = sunApparentSize;
  results(2,i) = innerPlanetApparentSize;
end
results'