Is there necessarily a formal definition of this field? What do you consider philosophy? Are certain subjects or studies more or less legitimate than others? Finally, what do you consider the purpose of engaging in philosophical discourse?

Philosophy is the definition of any subject, following logical principles. Ergo, anything that can coincide with this aspect, is philosophy. There are many subjects in philosophy, some of personal favorites are Pre-Socratic Philosophies, Aestheticism, Existentialism, Absurdism, Problem of Evil, Theological Studies, and any type of "philosophy" that deals with the definition of a "thing." Some philosophies are indeed more legitimate than others, especially when concerning other philosophies, such as anything dealing with a priori vs' a posteriori knowledge, especially when speaking for, or against, empirical observations.

Ethics, Politics, Art / Aesthetics, Metaphysics and Epistemology