"Men and women."
This sentence wants you to distinct between men and women, masculine and feminine, so the translation required is גבר, which stresses the gender of the person.
Anashim = אנשים = people. A nice man held the door for me = איש נחמד החזיק לי את הדלת. There are a lot of people in the streets tonight = יש הרבה אנשים ברחובות הלילה
you right, but i think a more accurate translation would be: person = איש - when its about a male person = אישה - when its a female peaple = אנשים - when its plural, and it doesnt referring to the gender
גבר is more about the gender. You use it when the gender of the person is important.