Is working as a radio director for an economic radio station that broadcasts stock market news permissible or forbidden, and is there any doubt concerning receiving a salary from a company that trades in securities?

It is not permissible to promote and market shares and transactions of prohibited stock exchanges because it entails leading to sin. If the broadcasting is free from that, then there is no harm in working with it.

As for working in a securities trading company, if the company is limited to permissible securities, there is no problem. But if it deals with all types, and you can limit yourself to the permissible ones, then the work and salary are not forbidden. Otherwise, it is not permissible to work in a job that directly involves trading prohibited securities. Any earnings from prohibited work included in the salary must be disposed of by giving them to public welfare or to the poor and needy.