Is it permissible for the master to examine his female slaves, and for the female slave to examine her master's male slaves? And do household utensils and furniture count as part of the inheritance? And how is gold distributed in an inheritance?

The master is not the slave, and therefore the question about him seeing his master's daughters is not applicable. However, if "master" is meant to refer to a slave, then many scholars hold that his master's daughters and wife are not required to veil themselves from him, based on the hadith from Aisha: "Enter, for you are a slave as long as a dirham remains due upon you." The Maliki school of thought permits a slave to look at the foreleg of his mistress and the hair of his master's wife.

As for a slave girl (amah), she is permitted to uncover her hair and extremities, as these are not considered awrah for her unless there is a fear of temptation (fitna).

The furniture of a deceased person's house is included in his estate and is divided among all his heirs. As for how gold is distributed in inheritance, it is distributed like other assets of the estate. If it can be physically divided, then it is divided; otherwise, its value is assessed and that value is distributed among the heirs.