Is it better for one who serves Islam (by writing a book, teaching Islamic knowledge, or engaging in a financial project) to not take any payment at all, in order to preserve sincerity, or to take money and then donate it or spend it on the interests of Muslims while preserving his sincerity?

The best of deeds is that which benefits Muslims the most. The greater its benefit to them, the greater its reward. So, if relinquishing wealth and giving it in charity is more beneficial to Muslims, then that is more preferable. And if taking possession of wealth and spending it on their welfare is more beneficial, then that is more preferable.