Can we calculate the orbit of exoplanets? I'm not an astronomer, but the question came up and I'm interested if we are able (or how accurate we are able) to calculate the orbital parameters of an exoplanet. Since the transit method must yield some properties of its orbit, we must know some of it. I also found some information here regarding the measurement of the orbital plane (Can we know the orbital planes of extraterrestrial planetary bodies?).

But do we have a clear picture of its orbit? Like we have of the planets in our system?

Could we, for example, tell if the planet was born in this system or has been caught by it at some point (giving it a different orbit I suppose)?
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 &amp; 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.

Some resources:

The HEASARC Exoplanet orbital parameter database

exoplanets.org