How can one propose to his cousin who lives abroad and posts immodest pictures on Facebook, knowing that the suitor cannot discuss the matter with his family or hers, and is concerned that she be convinced of him?

There is no impediment to choosing a woman of wealth. Religion should be the primary consideration in making the choice, for the Prophet, peace and blessings be upon him, said: "A woman is married for four things: for her wealth, for her lineage, for her beauty, and for her religion. So, succeed with the one who is religious, may your hands be soiled [with dust - an expression of encouragement]."

If your cousin proposes to the girl and she accepts him, it is not permissible for you to propose to her, for the Prophet, peace and blessings be upon him, said: "A man should not propose over the proposal of his brother until he marries her or he gives up." However, if she does not accept his proposal or she cancels it, there is no harm in you proposing to her.

Regarding the pictures the girl posted, she should be admonished by some of your female mahrams, or some of her male mahrams, while being cautious not to be lured by Satan.