Is a marriage in which the mahr (dowry) was not paid in advance considered void, and what is the ruling on a marriage in which the prompt mahr was not paid, but it was agreed upon to be paid upon divorce?

The questioner made a grave error by marrying a second time without ending the first marriage through divorce (talaq) or annulment (khul'). What the Sheikh did is neither a divorce nor an annulment, because divorce is solely in the hand of the husband, as the Prophet (peace be upon him) said: "Divorce is only for him who takes hold of the leg [i.e., the husband]." And annulment (khul') is a contract between the spouses. Her second marriage is invalid by scholarly consensus, as she was already married. Allah the Almighty says: (And [also forbidden are] married women except those your right hands possess) [An-Nisa 4:24]. The only way for a woman to be divorced against her husband's will is through a ruling by a Sharia judge to divorce her for harm (darar). It is obligatory for her to separate from the second man immediately. Her separation from him does not require annulment (faskh) or divorce (talaq) because the marriage is invalid, so divorce cannot be conceived therein. If she bore children from him, they are attributed to him due to the شبهة الزواج (semblance of marriage). After separating from the second man, she may return to her first husband, or request a divorce or annulment from him, then marry the second man with a new contract after her waiting period (iddah) ends. As for the argument that the first marriage was invalid due to not naming the dowry, it is weak. A marriage is valid even without naming the dowry originally, or by deferring it. If the dowry is not specified, the wife is entitled to the dowry of her peers (mahr al-mithl).