My main goal is to create games which is why I'm planning on studying IT so I can later on be a game developer/programmer.
My question is: does a game programmer get involved in the game designing process or is that only the game designer's job? Is a game designer always in needed or can the game programming team work on that at the same time?
Do programmers need to have game designing knowledge or do they just follow orders? Similarly, do game designers require programming knowledge? I'm kind of confused over what i want to be which is why I'm asking here.
Basically... do game programmers also get to be game designers?
It depends on the company. Roles and titles vary widely across the industry, so at some companies roles will be very rigid and strict and at others they will be more flexible and allow for more cross-discipline work.
It also depends on the person; some developers like to explore beyond their "technical" role or job title, others don't.
You'll probably find more flexibility in a smaller game company, but it's certainly possible to find the role flexibility you are hoping for in a larger one. You just need to be aware that's something you want and look for it during the interview process. Or start your own company where you can be everything and do everything.
A title is just a title so it will vary from company to company, but usually a game designer does a lot of scripting of events/quests etc. and if you are hired on as a straight up programmer it is very unlikely that you will have any input into the game design.
In short being a programmer for a game company is not much different from other places, while being a game designer means you will design games but will usually also have to know a decent amount about coding or at least scripting.
As the other answers have pointed out, this will vary from company to company, but I need to point out something about the video game industry, something that it has in common with all other branches of the entertainment industry.
In short, game design is the "fun" part, so the people in charge, and the people that have paid their dues, tend to monopolize it. People with less experience, or less seniority, tend to see their attempted contributions dismissed out of hand.
Compounding this, so many people think they want to work in the video-game industry, and the people in charge know this, so they have little motivation to treat newcomers kindly, since they are so easily replaced. So don't expect to successfully call them out on their exclusive behavior.
There are other nasty truths in the video-game industry, such as long hours, emotionally immature peers, and video-game consoles that are toys in more than one sense of the word, but they're beyond the scope of an answer to your question. The bottom line for me is, I'm glad to no longer work in the video-game industry, and never want to go back.