Is it permissible to propose to a girl I love, knowing that I cannot be without her, and that her mother engages in unacceptable behavior, talking and going out with young men, but the girl herself is not convinced by her mother's actions?

There is no impediment to proposing to and marrying a girl if she has repented, is chaste, and adheres to the Sharia, even if her mother is not observant, as long as the girl disavows her mother's actions. It is preferable to seek out a religious family due to the positive effects that result from such a choice.