Hi Kristintrapp!
Happy to try to help bring some clarity to this one. Let's start with unpacking (E): "It is an example that is offered in support of a premise that is intended to support the argument’s main conclusion directly."
The "example" in (E) is "the proposition that certain entrepreneurs fail in managing company growth," which is used in support of the premise that many entrepreneurs who succeed in starting a business end up failing for lack of managerial skills. And that, in turn, is used to support the main conclusion, starting at "Hence,..."
To your first question,
My question is this: would it be acceptable to say that examples and analogies are not premises, and instead, they are supporting details that are used only to support premises? Could I have eliminated D. simply because it didn't explicitly state that it was an example?
I think the problem with (D) is the "directly" language--the example is evidence that one of the premises is true, so it's directly supporting a premise, but not the main conclusion as (D) suggests.
I think that relates to your second:
I'm just trying to understand if in the future I can eliminate answers that don't state an analogy/example as an analogy/example. Can they also be called premises in themselves?
If I understand your first question correctly, even if "premise" was changed to "example," (D) would still not be an accurate description because of the "directly" language. At the same time, to your second question, a correct answer choice could certainly refer to an analogy/example as a premise, if that's how it's functioning in the stimulus. Further, the stimulus might even label a sentence as an analogy--it might be phrased, "Analogously,...", and an answer choice might appropriately refer to that analogy as an "example," if it is indeed functioning as an example. So "analogies," "examples," and "premises" are all distinct, though a sentence could certainly be all three of those, or two of them, or just one.
Hope that helps!