A proper noun is usually understood as a noun that is capitalized. In order to be a proper noun, like any other noun, the word must be a person, place, thing, or idea.
Proper nouns, no matter where they are in the sentence, have their first letter capitalized.
When at the beginning of a sentence, it does not matter whether or not the noun is proper: it is capitalized anyway. This is exactly why you have to learn exactly which words are proper nouns by memorization.
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There is no “hate speech” exception to the First Amendment. Contrary to a common misconception, most expression one might identify as “hate speech” is protected by the First Amendment and cannot lawfully be censored, punished, or unduly burdened by the government — including public colleges and universities.
