Why is research important in the groundwork of our writing, be it fiction or non-fiction?

The best writing, particularly in fiction, comes from understanding the details we might throw into a story as well as the overarching histories/cultures we might come across. Tossing in random elements because they might sound good without understanding what cultural/historical contexts they come from proves us to be failures as writers.

Why should we be careful about projecting our own cultural mores on the past?

In studying history we need to understand it within its own context. While it may be interesting/pertinent to study how it reflects upon us in the present, your focus should be on how it reflects upon its own past and the changes that led up to that moment in time.

Why is it important to study how culture is transmitted across a physical landscape as well as a temporal one?

In understanding cultural transmission across space as well as time, we can learn much of how humans as a whole (as well as on a smaller scale, two or more groups) interact. Culture does not appear in a vacuum, but is instead shared and learned through interactions.