Is it permissible for a Muslim to conduct tests to ascertain the true nature of the person before him, such as leaving money unattended to gauge the other's trustworthiness, or sending the mother during engagement to check the cleanliness of the suitor's or fiancée's family?

It is not permissible to test one's wife and children with actions like those mentioned, because the default is to assume the good intentions of Muslims, and it is not permissible to search for their mistakes, as this involves the ill-suspicion (zann) that is forbidden by the Almighty's saying: "Avoid much [negative] suspicion, indeed some suspicion is sin." The Shari'ah has also forbidden prying into private matters, for whoever pries into them, Allah will pry into his private matters and expose him.

Nor should the fiancée's family be surprised with an unannounced visit, as this can cause them embarrassment. They might have other engagements, or the house might not be tidy. The Shari'ah permits returning if entry is not granted, and this is "purer for you." Knowing the circumstances of the fiancée and her family is best and safest done by asking trustworthy individuals about them.