Is it permissible for the husband to donate to his wife some of the expenses for Hajj, and not demand them back from her, hoping for reward from Allah, knowing that she is employed and can pay them back later?

Every good deed a husband offers his wife, he will be rewarded for it if he intends thereby the face of Allah, whether it is paying for her Hajj expenses, or part of them, or medical examination costs, even if the wife is wealthy. This is based on the Almighty's saying: "And never will He deprive you of [the reward of] your deeds," and His saying: "So whoever does an atom's weight of good will see it." This is not considered a mistake or a waste of money; rather, it is a reward and a sign of good companionship, in confirmation of the Prophet's (peace be upon him) saying: "And indeed, whatever you spend of expenditure, it is a charity, even the morsel you raise to your wife's mouth."