Is it permissible for a woman or anyone to say to another, "by the right of my love for you," to ask for something to be done or left undone, and is it permissible to say, "I implore you," for the same purpose?

Swearing by other than Allah is forbidden because the Prophet, peace and blessings be upon him, prohibited it. So, whoever swears an oath, let him swear by Allah or remain silent. As for making the husband's love a means to achieve a desired outcome, or appealing by kinship and interceding through family ties, there is no harm in it if it is in the sense of hope and persistence in a request, and seeking help through emotional appeal.