Is it permissible for a brother who owns an Islamic website that contains paid advertisements to personally benefit from a portion of the profits generated by these advertisements, or must all profits be used solely to support the Islamic website?

There is no objection to taking financial compensation for publishing advertisements, provided that what is displayed is controlled to avoid promoting something objectionable or forbidden. As for personal benefit from the profits, it is subject to the ownership of the website, so there is no objection for the website owner to benefit from them, even if the website is Islamic.