For example, if a woman were to have a beard, would she be permitted to shave it off with a razor or otherwise destroy it, or do these restrictions apply to both men and women?
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Nope. It does not apply to women.
Source: Mishnah in Kiddushin (1:7)
According to Rabbi Reuven Lauffer at Ohr Somayach (writing in 1999 in response to a question about Omer restrictions), no: prohibitions against removing facial hair do not apply to women.
I've seen the question and answer from the person who asked it. The question asked about Omer restrictions; the response was:
I don't know what it means when a response doesn't reiterate specifics from the question -- that is, whether he is speaking broadly or only about the Omer. (The answer did not cite sources.)