Is it permissible for my father to pay off his debts and cover his household expenses with money he collects from people, claiming there is a debtor (or other cases such as an accident or an elderly woman), without revealing his true identity as the debtor?

The agent is a trustee of what is under his hand; it is not permissible for him to violate the condition set by the principal. If the donor stipulates that the agent should not spend from the charity on himself, then he is not permitted to do so.

However, if no such condition was stipulated, and the donor merely intended for the charity to reach any needy person, then scholars have differed regarding the permissibility of the agent taking from it for himself. The most cautious and absolving approach for one's conscience is not to take anything from it unless the donors are informed of the possibility of him taking for himself.

Whatever is collected for a specific category, such as "an elderly woman" or "a young man who had an accident," this specific category must be adhered to. Otherwise, the money must be returned to the donor or their permission sought to spend it on a category other than the specified one.