Is the husband entitled to force his wife to relinquish all her rights, including child custody and the full mahr, in exchange for not defaming her, after a strange man was found in her house during the husband's absence, and the wife claimed that the man was asking about a friend of hers and disappeared for fear of her mother-in-law?

The basic principle is that a Muslim is free from sins, and it is not permissible to judge them based solely on suspicion and doubt. What is required of a Muslim is to have good thoughts and to conceal the faults of their brother. It is not permissible for a man to defame his wife or force her to relinquish her rights in exchange for not defaming her, for her rights do not cease except with her consent and willingness; otherwise, it would be consuming her wealth unjustly.

However, if it is proven that the wife is involved in an illicit relationship, it is permissible for the husband to put pressure on her until she ransoms herself through khul' (divorce at the wife's instance), based on the Almighty's saying: "And do not constrain them [forcibly] to take away part of what you gave them, unless they commit a clear immorality."