Does the husband have the right to demand the mahr and the upbringing of the child after discovering that his wife had relationships with young men before and after marriage, and after she falsely swore upon the Holy Qur'an?

The woman became your wife through the contract, and by your consummation of the marriage, she became entitled to the full dowry. You do not have the right to take it from her due to her emotional relationship with a non-mahram. However, if she does not repent, you have the right to make things difficult for her until she ransoms herself from you with a compensation that you both agree upon in exchange for divorce. This compensation could be her waiving the dowry or the custody of the child.