Is what happened between the questioner and the girl considered Zina (fornication/adultery), and will Allah forgive them after repentance, and what should they do to please Allah, and will marrying her please Allah despite the mother's disapproval for an unconvincing reason?

You both must hasten to repent to Allah for the sin of adultery you have committed, which is one of the gravest sins. Repentance entails abandoning the sin, regretting it, and resolving never to return to it. It is not permissible for a man who has committed adultery with a woman to marry her until after repentance and the expiration of her waiting period (iddah). If you both repent to Allah with sincere repentance, then there is no blame upon you for marrying her, unless your mother forbids you from marrying her for a valid reason, in which case you must obey her. However, if she forbids you for no valid reason, then there is no blame upon you.