The assumption is made that the planets of our solar system all have elliptical orbits elongated in the same direction. Further, generalized to that orbits around any object with its own orbital vector should have elliptical orbits, including moons orbiting planets.
Actual diagrams, rather than made-up artists' impressions, can be found at NASA JPL. https://ssd.jpl.nasa.gov/?orbits
The ellipticity of planetary orbits are in general quite small and they are not aligned. http://www.met.rdg.ac.uk/~ross/Astronomy/Planets.html gives a table of orbital elements in the solar system. The longitudes of perihelion (labelled as "$\sim \omega$") are different - indicating that the "eccentricity vectors" (as you put it, and keep asking for) are pointing in essentially random directions around the ecliptic plane.
Many of the diagrams that you see are drawn from perspectives that are not looking straight down on the ecliptic plane. If that is the case then of course you see (highly) elliptical shapes (even when the orbits are nearly circular) that appear aligned. That is merely a perspective effect.
First of all, to make absolutely sure we're on the same page about the shape of the elliptical orbits, seeing as there's been some confusion in the comments, I've included the same two pictures from both top down and at an oblique angle.
Which directions are the planets' orbital eccentricities pointing to?
Mercury is a bit of an anomaly because its orbit precesses over time and so the direction of its major axis changes very slowly (it takes 12 million orbits for the major axis to do a full 360 degree sweep). The other planets have their major axes scattered fairly evenly, by my eye.
Which direction is the sun moving in?
This was answered here. The Sun is moving through the galaxy in a direction 60 degrees from the ecliptic, which is not aligned with the eccentricities of any planets.
No, they do so because of the impulse of the cloud the solar system formed from. This impulse is preserved in the orbits of the solar system bodies, and in their rotations.
It is a dynamic system with dependencies and influences through gravity. Objects can get captured, flung around, forced into resonances ...
The orbital plane ("ecliptic") is different for all objects. Differences may be small. The solar system's ecliptic is defined by earth's orbital plane mostly for [nautical reasons] . Other planets and objects have an inclinitation in relation to that plane.
Further links that might help: