Is it permissible, Islamically, for a girl to speak with her paternal cousin—who is of good character—within narrow limits, to ascertain his desire to marry her, given her inability to inquire through her parents or the women of her family?

There is no need to speak with your cousin. Instead, ask your father to allow the suitor to visit you to express your opinion. If your cousin is a man of good character and religious commitment, and you wish to marry him, then it is permissible for you to speak with him and ask him to propose to you, provided that you adhere to the Sharia guidelines and that the conversation is limited to what is necessary. You may also stipulate that he keep the matter confidential.