Is it permissible to promise a girl marriage without her family's knowledge, and is this considered among the whispers of Satan that open doors for him, and what is the best according to Sharia to avoid error in such a situation?

There is no harm in informing the girl of the desire to marry her, in order to know her acceptance, and to ask her to wait until one is able to afford the expenses of marriage, provided that the limits and etiquette of Sharia are observed, and caution is exercised against temptation. It is permissible for a woman to propose herself for marriage, even if she has guardians.