Is a Gentile housekeeper permitted to drink out of the same kiddush cup as the rest of the family? If not , why not?

If the wine is mevushal, cooked there is no problem at all. If the wine is not mevushal, and she drinks all that is left in the cup, again no problem. If she does not finish the cup, there might be a problem of mukzah on the cup- see Beitzah 21b. It is unclear what Rabbi Yehoshua Ben Levi holds in this instance.