Urban Core, Urban, SubUrban, Rural Definitions

Unless you are dealing with definition of zones for light pollution in SSc8, I don't think you have to worry about being very precise. You could check the zoning map and its zoning designations - these often have wording that gives you a sense of where you fall.

It depends on the particular region, but if I had to throw out a number I'd say 7-10 dwelling units per acre could be urban, less might be considered suburban. (I know a New Yorker who thought that a "city" meant only apartment buildings or row houses, and no single family detached homes...!)

I usually go with my intuition on the difference between Urban and Suburban, where Suburban implies you can't realistically walk to your surroundings. For example, I work in a Suburban office environment where our team typically drives to lunch, but I live in an Urban neighborhood where our family walks/bikes to dinner. Think "suburbs" and the level of connectivity available. I also agree with David's suggestion to check the zoning designation in case your jurisdiction already defines your project site in one of those categories.

Is a university residence hall on a college campus with other student housing throughout considered urban?

Rachel, I would look at whether the college campus is an urban/suburban/rural campus. For example NYU would be considered an urban campus (in Manhattan), but a school set in the country (thinking of Clemson University in SC) could be suburban or maybe even rural depending on where that dorm was positioned.