Is the aforementioned act considered unlawful (haram)? Is the girl considered a legitimate wife? And will she be his wife in Paradise if they both enter it?

Since you have entered into a valid marriage contract with the woman, she has become your wife, and it is permissible for you to consummate the marriage with her, even if the customary wedding ceremony has not taken place. However, it is preferable to delay consummation until after the wedding. You have not committed anything forbidden by consummating the marriage with your wife. It has been reported in the Prophetic Sunnah that a woman will be in Paradise with her last husband from this world.