Is it permissible to quote writings from an author whose name is unknown, and to listen to anasheed downloaded from YouTube, with doubts about the legality of the accounts that uploaded them, and is it obligatory to delete what was previously downloaded?

There is no requirement to seek permission when quoting from the works of others; rather, it is obligatory to attribute statements to their original sources if they are quoted verbatim. If the author's name is unknown, it suffices to cite the source or state that it is a نقل (transcription or نقل) reproduction.

As for listening to copyrighted content without publishing it, the matter is lenient, especially if it has not been definitively established that the accounts are illegal. Some scholars permit copying for personal benefit, provided there is no commercial gain from it.