Is it permissible for my father to go back on the promise he made to me to provide housing, which was the basis of the marriage agreement? And is this promise considered a gift that is not binding upon him until it is taken possession of, or is it a binding promise?

It is not permissible for a parent to gift one of their children without gifting the others; equality must be observed between them in gifting.

If there is a valid reason for favoritism, then gifting the usufruct (right of dwelling) is sufficient without transferring ownership of the property. The property remains in the father's name and returns to the estate after his death.

As for the promise, it is religiously binding unless there is an excuse, and legally binding if it is contingent upon a cause and the promisee incurred expenses because of it. Fulfillment is obligatory, or compensation for damages.

You can try to convince your father to allow you to dwell in the property without transferring ownership, and be diligent in showing him dutifulness even if he falls short in his duty towards you.