Is it permissible to work for financial compensation on forums similar to "star-dima.com/vb", where the income is derived from Google AdSense and other sources whose permissibility is questionable, knowing that the work will be in obedience to Allah and for the benefit of Muslim youth, and that it will increase the profits of the company owner?

If the field of work is permissible, then there is no harm in it, and it is permissible to benefit from its earnings. It is not affected by the fact that the website's capital might contain an element of doubt regarding being unlawful, as long as the transaction itself is permissible; this is due to the permissibility of dealing with someone whose wealth is mixed [with lawful and unlawful sources].