The characteristics of the orbit of an exoplanet can be determined very accurately using measurements of the reflex motion (the "Doppler wobble") of its parent star. These measurements yield the orbital period and eccentricity of the orbit. If we can estimate the mass of the parent star, then the orbital semi-major axis is known through Kepler's 3rd law. Orbital periods, and hence semi-major axes, can also be measured very precisely from transit data, and eccentricities can also be estimated provided one has a good idea of the mass and size of the parent star (e.g. van Eylen & Albrecht 2015).
Additional details can be gleaned from transiting exoplanets, such as the inclination of the orbit to the line of sight and any difference between the orbital axis and the spin axis of the star it orbits. These measurements exploit something called the "Rossiter-McLaughlin effect", as explained in an answer to the question you linked to.
The answer to your broader question is yes, of course people are looking at the distribution of orbital parameters: the behaviour of eccentricities versus planet mass and semi-major axis; the prevalence of planets of particular mass at certain distances around different types of stars; the frequency of occurence of multiple planets and orbital resonances, the misalignment between the spin of a star and the orbits of its planets, and so-on, in order to try and understand how planetary systems form and evolve.
The HEASARC Exoplanet orbital parameter database