This thesis explores selected questions in several active areas of research in star and planet formation. Chapter 2 constrains the star formation history and initial mass function of the old stellar population in the Galactic Center nuclear star cluster. Preliminary results suggest an initial mass function for this region that differs significantly from the standard mass function observed in other parts of the Galaxy. Chapter 3 derives stellar and planetary properties for an extraordinary planetary system that surrounds an unusually low-mass star and includes the closest, smallest, and least massive planet known to date. The data presented in this chapter were valuable in enabling a follow-up transit detection of this planet that placed strong constraints on its internal structure and composition. Chapters 4 and 5 analyze observations of extrasolar debris disks in an effort to infer global characteristics of the underlying planetary systems, including the location and properties of planetesimals and planets. Specifically, Chapter 4 presents multi-wavelength imaging observations of a debris disk that potentially point to the existence of an unseen exoplanet. In contrast, Chapter 5 considers substructure in a different debris disk that is unlikely to be gravitational in origin. The results suggest that debris disk interactions with the interstellar medium may create asymmetries that mimic the presence of an exoplanet at short, scattered-light wavelengths
