Is it permissible for me to copy free Islamic material from various books and websites that do not contain copyrights, without mentioning their source, knowing that I do not attribute it to myself and that some of these sources may contain Sharia violations such as music and pictures of unveiled women?

There is no harm in benefiting from the Internet and transmitting useful articles, provided two conditions are met: attributing the articles to their authors (even if by pseudonyms), and not altering the content in a way that contradicts the author's intent. If the original website is not suitable for publicizing, it is sufficient to state that the article is transmitted from the Internet. One must be cautious about taking someone else's work without notification, as this is a breach of trust.