Must the questioner believe the wife's claim of pregnancy despite no penetration, and does handing over the deferred mahr without divorce constitute a divorce, and does the noble verse apply to it: {O you who have believed, when you marry believing women and then divorce them before you have touched them, then there is not for you any waiting period to count against them. So provide for them and release them with a gracious release}?

What you have embarked upon is a reprehensible and blameworthy marriage because you married a woman without verifying her eligibility for marriage. She might have been married, or in her waiting period (iddah (definition)), or a prostitute, and the blame falls upon you. The marriage contract is not valid if there are female relatives who have a greater right to give her in marriage, such as the father, then the grandfather, then the son, then the full brother. This marriage is considered a شبهة (doubtful) marriage, and its effects, such as attributing the child and inheritance, are established. Marriage is not a game. Divorce does not occur merely by paying the deferred dower but by uttering the divorce pronouncement. The pregnancy is attributed to you unless its attribution is impossible or you deny it through li'an (oath of condemnation). It is not permissible to accuse a woman without evidence. The mentioned verse does not apply to you because you have secluded yourself with her. If you divorce her, she is entitled to her waiting period (iddah) and the full dower.