Is it permissible for the husband to spend on his wife's daughter from a previous husband, knowing that her father is alive and financially capable, but the daughter lives with her mother?

The child's maintenance is the responsibility of his father. The husband is not obligated to financially support his wife's daughter from another man. However, if he spends voluntarily, he will be rewarded. And if he spends due to her father's refusal to provide maintenance and intends to seek reimbursement from him, he is entitled to do so.