Is marriage to a woman who was previously an adulteress valid, and does the religion necessitate remorse for the sinner? And must she be divorced due to an inability to bear what she did, especially with his feeling of the sincerity of her repentance and his fear for her of perdition, along with her insistence on leaving his first wife and children?

Regret is a fundamental pillar of repentance. It is authentically reported from the Prophet, peace and blessings be upon him, that "Regret is repentance." Since you have perceived the sincerity of her repentance, strive to forget the past and do not divorce her. It is not necessary for her to repeat the sin or for the breakup of her family to affect her. Be keen on nurturing her in faith, encourage her to attend gatherings of knowledge and goodness, and connect her with righteous sisters, for a person is influenced by their friend. Be a good role model for her, and reject her request to leave your wife and children. Divorce is permissible but disliked unless there is a need, and the default is prohibition except for a necessary extent.