I just have started answering questions. My question is: if a question has many answers and has an accepted answer, should I answer? I am afraid my answer will receive lots of down-votes.
You should answer the question if you can add something useful. Maybe you even want to elaborate on a short but correct answer. Then you could refer to that answer and elaborate, maybe show some proof, other solutions, etc.
It is fine to add new answers to accepted questions. That's why there is still a Post Your Answer button.
You do want to be careful though to add quality, not just a single comment-like sentence that doesn't add value to the existing answers. Such posts will get down-voted.
I think that it's important to remember that acceptance is by no means permanent.
If you are able to post a fantastic answer, outshining all the other answers, the question's author has the ability to un-accept the accepted answer and accept yours.
Even if that doesn't happen, you are still able to receive up votes. There are a number of questions where the accepted answer has less score than another answer. (Accepted answers may even be highly down-voted!)