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?
Nope. It does not apply to women.
Source: Mishnah in Kiddushin (1:7)
וכל מצוה בלא תעשה,
בין שהזמן גרמה ובין שלא הזמן גרמה,
אחד אנשים ואחד נשים, חייבין;
חוץ מבל תקיף, ומבל תשחית, ומבל תיטמא למתים.
And every commandment which is a prohibition,
whether it is time-dependent or is not time-dependent,
both men and women are obligated;
except for "You shall not round off [the corners of your head],"
"You shall not destroy [the corners of your beard]"
and "You shall not become ritually impure for the dead."
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:
There is no prohibition for a woman to shave or remove facial or body hair. The prohibition against removing facial or body hair only applies to men.
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.)