For short period massive planets, orbital eccentricities can be routinely measured using radial velocity observations, while observing the Doppler shift during a planetary transit can constrain the star-planet obliquity (known as the Rossiter-McLaughlin effect).
I focus on small planets on longer-period orbits, for which such radial velocity observations are not possible. Studying stellar pulsations can provide the inclination of stars independent of the size of the planet, while knowing mass and radius of the planet can provide the orbital eccentricity by careful analysis of the planetary transit.
When space missions (e.g. Kepler) observe a very tiny dip in brightness (of only a few hundred parts per million), we hope the cause is a transiting exoplanet moving in front of the star, blocking a tiny fraction of starlight. Binary stars can eclipse one another and also cause transits, but those eclipses are typically much deeper.
However, there are many stars in the sky. Sometimes several are in the same line of sight, so faint background stars can contaminate observations. If a faint binary star contaminates the bright star we observe, a deep binary star eclipse might be heavily diluted and look like a planetary transit. What we believe we see is a planetary transit, but instead we are observing a binary star in the background. These are false positives.
False positives can be ruled using follow-up measurements and additional observations, e.g. radial velocity follow-up or multi-colour photometry. High-resolution images from ground-based telescopes can spatially separate stars and reveal contaminant stars. I am particularly interested in using the shape of transits to rule out false positives, as well as combining optical and infrared photometry.
Analyzing phase curves, as well as secondary eclipses when the planet disappears behind its star, we can estimate the temperature of the planet's permanent dayside as well as its permanent nightside. Furthermore, the reflective properties (albedo) of the planet can be measured. The relative contribution of the planet's own heat and the reflection of light from the star depends on the wavelength, making it crucial to use multi-color observations.
Phase curves thereby provide information on the planet's atmosphere, including the presence of possible clouds.
Using asteroseismology, we can obtain precise stellar parameters, such as the mass, radius and age of the star. In some cases, asteroseismology can even be used to measure the stellar inclination angle and rotation period.