Is what the girl did by kissing her fiancé before the wedding and being intimate with him, leaving a mark that sparked gossip, forbidden? And will her mother's supplication against her be answered after punishing her?

It appears that the intended meaning of "كتب الكتاب" (katab al-kitāb) is the legitimate marriage contract. If that is the case, then he is your husband, and there is no harm in kissing him. Your mother's cursing and praying against you are baseless. However, it is advisable to observe the custom of not being in seclusion until the wedding to avoid embarrassing the family.