Is it permissible to pay a person to design a free blog on "Blogger" dedicated to attracting customers for companies and websites that profit from advertisements, and is it permissible to take payment from these companies if their capital is mixed with illicit funds due to advertisements for prohibited things that users do not see?

It is permissible to profit from companies and websites if the dealings are lawful. It is also permissible to pay money for designing or programming a blog to attract customers, provided that the advertising is restricted to lawful dealings. Similarly, it is permissible to watch lawful advertisements and receive payment for them, even if the companies' funds are mixed.