Yes, we do. It is one of many words and expressions that have found their way into ordinary English. When used in the United States, it is often pronounced "bone ap-ə-teet" (i.e mispronounced); "appetit" is (mis)pronounced like the English word "appetite" with "-tite" changed to "-teet." The phrase was always popular, but the late Julia Child was the author of famous book called "Mastering the Art of French Cooking," and had a very popular TV show, and she always closed by saying "bon appetit," which made it even more popular. Other French phrases used in US English include "bon voyage," "cherchez la femme," "laissez-faire," "RSVP" on invitations (an initialism for "respondez si vou plait"). Of course, English includes literally hundreds, probably thousands of French loan words.

Americans do use Bon Appetit. Some say "Enjoy" or "Enjoy the meal". Religious households pray to thank the lord for the food he had graciously given before eating.

Yes. The English language uses a lot of French phrases and words.