Is the monthly expenditure on the wife's mother considered a charity for which the doer is rewarded, if he intends it as charity and to bring happiness to his wife, given that the wife's mother is a widow and poor?

As long as your wife's mother is a poor widow, your spending on her is an act of charity (sadaqah) and kindness to your wife. If you seek the countenance of Allah through it, then it is a righteous deed, among the best of deeds, for the Prophet (peace and blessings be upon him) said: "The one who strives for the widow and the poor person is like the one who strives in the way of Allah, or the one who stands (in prayer) at night and fasts during the day."