Is it permissible for me to secretly marry a married man, knowing that I will marry myself without anyone's intervention, and he will provide me with a house that he will regularly visit for a certain period of time? And is an urfi marriage to him permissible?

A woman is not permitted to marry without the permission of her guardian (wali), because the guardian is a fundamental condition for the validity of the marriage contract (nikah). If the marriage takes place without the guardian's permission and without two just witnesses, it is invalid. As for publicizing and announcing the marriage, it is desirable, and its absence does not affect the validity of the marriage, though announcement is preferable. If the marriage is for a specified period after which it ends, then it is a void temporary marriage (mut'ah), even if all other conditions are met.