Is it possible to have an addiction to the internet?

Wording choice - yes. There is a difference between compulsion and addiction - so yes some people feel compeled initially and if left unchecked can become addicted.
It is. Internet addiction is the subject of new research. "behavioral addictions" altogether (e.g., gambling, pornography, sex...) is relatively new and remains the topic of some debate. But imo it is clear that behavioral addictions, including to the internet and/or its content can become severely damaging to people's lives. In my experience it is best dealt with by an addiction expert. Ask your md.
I suppose. If that is all you do or it causes harm to yourself or others, it's possible. I suspect what you have is a preoccupation which can be harmful to a lesser extent. You probably need to see a psychologist to explore this issue if you are suspicious of this being a problem.