Is it permissible to accept a reward (laptop) from a website that contains both permissible and forbidden sections (movies and songs), knowing that my participation is limited to the permissible topics?

This reward falls under the category of ju'alah (a contract for a reward). A condition for its validity is that the work must be permissible. So, if the questioner's contributions to the website are permissible, then there is no objection to her taking the ju'alah; otherwise, there is. The mixing of permissible and impermissible elements on the website does not affect the permissibility of the ju'alah, although one should consider the benefits and harms resulting from participating in such websites.