The small-body populations of the distant Solar System inform our understanding of the structure, formation, and evolution of the Solar System. The orbits of these Trans-Neptunian Objects (TNOs) act as tracers for dynamical activities either ongoing or past. The distributions of TNO sizes are set by, and so probe, the conditions of the formation and evolution of the Solar System. Using data from surveys on the Canada-France-Hawaii Telescope, I constrain the size distribution of a TNO subpopulation: the scattering TNOs. The scattering TNOs are chosen as they have orbits that come in closer to the Sun, therefore allowing smaller TNOs to be detected. The characteristics of size distribution for the small-sized TNOs is an important, and only recently observable, constraint on the formation of this population. I find that the H-distribution is consistent with models where TNOs form as large (50 km - 100 km) aggregates from the proto-planetary nebula. A recent discovery of apparent clustering in the orbits of some TNOs has led to the hypothesis of an additional and unseen planet in the distant Solar System. I examine the formation implications and consequences of such a planet, and the biases in the detected sample used to infer the planet's existence. Via a combination of dynamical simulations, survey simulations, and statistical comparisons of the observed TNOs, I explore the additional planet hypothesis to determine if there exists strong evidence for an additional planet in our Solar System. I find that there is currently no strong evidence for the clustering of orbits in the observed sample and that the proposed additional planet does not produce such a signature in models. / Graduate