Learn Russian: Я должен написать сорок три слова.

In Russian compound numbers (like 21, 43, 154), the case of the noun is determined only by the very last word in the number. Since the number ends in три (three), it acts just like the number 3 on its own, which requires the genitive singular form: слова.

Why doesn't сорок (forty) look like the other "tens" words, like пятьдесят?

The word сорок is a historical exception! Instead of being built from "four" and "ten", it comes from an old Russian word for a bundle of forty animal skins used in trade. You just have to memorize this one as the special word for forty.

If a woman is speaking, does the word должен change?

Yes! Должен is a short adjective, so it has to agree with the gender of the person who "must" do something. A man says Я должен, but a woman would say Я должна.

Why use the perfective verb написать here instead of the imperfective писать?

Because you are talking about completing a specific, one-time action with a clear result: getting exactly forty-three words written down. If you were talking about the continuous process of writing, you would use писать.