Thank you for the answers.
So the tasks of a game designer in general would be content, world, system design and story writing.
In general in medium and larger companies do game designers need to know how to program?
System design, yes. Content and story writing, not so much. Content is the art, text, and music that actually go into the game, and in a larger team these details will be delegated to an artist, writer, or musician as appropriate. Only in a small team is the designer a primary content creator. Instead, besides system design, major tasks of a designer include concepting (which is like writing outlines/descriptions or sketching needed content, rather than creating the actual content) and documentation writing (the features list, design document, flowcharts or a storyboard). Good communication skills are also used for writing lots of emails/posts and assignment descriptions to any other team members; and let me again mention research, it's actually important and requires research and analysis skills that many people don't have.
If the designer is also handling the project manager role, they would also have to do things like writing recruitment materials, conducting interviews, creating a project website or private forum, creating a kickstarter, or presenting a proposal to investors; all sorts of scheduling, maintaining team morale by giving pep talks and praise, and arbitrating between arguing team members are also part of the manager's role. The project manager role takes a ton of leadership and secretarial skills.