Is it permissible for a person to use a talent discovered by using neighbors' internet without their knowledge, and can he earn a living and write books with this talent, with a legitimate Islamic loophole such as giving charity on behalf of the internet owners?

If using the internet causes harm to others, then it is a sin for which one must seek forgiveness. Giving charity with the intention of expiating for this violated right is a good deed. There is no harm in benefiting from the writing and authorship talent that has emerged, but it is not advisable for its domain to be on the internet due to its many pitfalls. It is recommended to direct this talent towards writing beneficial رسائل (epistles/treatises) or books, focusing on calling to Allah, enjoining good and forbidding evil, and teaching beneficial books, with the primary concern being to write what is useful and draws one closer to Allah.