Is it permissible for a man to propose to the daughter of the daughter of a woman with whom he had "direct intercourse from behind" and touched her vagina with his hand, while ignorant at the time and having repented?

What you did was an ugly act, and you have committed a great sin if you were an adult. You must hasten to repent and seek forgiveness. This act is not considered zinā (fornication/adultery) in the sense that incurs the prescribed punishment (حد), nor does it prohibit you from marrying the female relatives of that woman according to the majority of scholars. Therefore, you are permitted to marry the aforementioned girl.