Is the husband entitled to demand money or the like if his wife requests a divorce from him, knowing that he does not pray and did not ask about her when she was pregnant? And is the wife liable to pay rights if she testifies in court that he does not pray?

If the husband abandons prayer, then if he is considered an apostate by some scholars, separation between them becomes obligatory through annulment, not divorce, and the husband is not entitled to any money. If he is not considered an apostate, then it is permissible to seek divorce due to harm, and there is no obligation to pay the husband money if the judge rules so after his abandonment of prayer is proven. If there is no proof, then one can seek divorce or khul’ (divorce at the wife’s instance, usually with compensation) in exchange for money. One should try to advise the husband and seek help from those who can assist him in finding guidance. If he becomes righteous, that is good; otherwise, do not hesitate to seek a divorce.