Is it permissible to transmit or quote fatwas, sayings, hadiths, and legal opinions from websites or shaykhs for the purpose of da'wah and advising others, and in this case, is it necessary to attribute every piece of information to its original source?

There is no objection to transmitting information from scholars' lectures or websites without attribution if the information is fragmented and compiled. However, if an entire fatwa is quoted, the source should be mentioned. It is advisable to continue inviting people to Islam and advising them, and not to pay attention to doubts.