Is a wife's grumbling and her saying, "You forbid me from remembering Allah, and this is unlawful for you," actually unlawful when her domestic duties are requested of her, and do Qur'an recitation sessions at work make the salary received unlawful?

It is permissible to recite the Qur’an during work hours in times of leisure, but it is not permissible if it distracts one from work. A woman's attending to her husband's and household affairs is better than her being preoccupied with memorizing the Qur’an, and she may incur sin if she neglects her marital duties because of it. You should rejoice in your husband's eagerness to memorize the Qur’an and assist him, and emulate the Messenger of Allah, peace and blessings be upon him, in serving his family.