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If I am briefly paraphrasing a famous figure's quote, or summarizing a certain figure's thesis on a certain subject, should I cite some source? These ideas have just kind of stuck in my head from college or independent research. I didn't research them just to stick them in here.

I suppose my options are:

1. Remove the references totally. They are pretentious and unnecessary, and offensive to the word limit.
2. Footnotes/works cited.
3. Don't cite. You aren't quoting directly.

I don't want to flout the word limit, but I think these references are relevant (kind of like how test prep companies would tell you to reference a work of literature on the SAT writing section to demonstrate breadth). But I also don't want to plagiarize...

I've been bluebooking every piece of my writing the last 15 years. I'm surprised you haven't been doing the same. I see Cooley in your future.

I don't really have any authority on this, but intuitively I would say that as long as you're referencing the famous figure/work directly in the body, there's no need for a footnote/citation. As for offending the word limit, etc, I'd only be worried about that if the quote is particularly trite/pretentious. But a well chosen quote should be fine.