Is it permissible to propose to a girl after she has been engaged to another man, and then her family offers her to him again, knowing that he did not propose to her initially, and that the rejection came from the girl's side before the first suitor's rejection?

There is no harm for a man to propose to a woman if she did not accept the first suitor. The ruling of her guardian's acceptance is nullified if she dislikes him, so she has the right to retract her acceptance, or if the suitor withdraws his proposal, then it is permissible to propose to her.