eye contact is definitely an important aspect in terms of building a rapport with your interviewer, showing that you are attentive, etc. However,I would say it is not a dealbreaker unless you are super awkward.
To your first question, one mistake I see people make is looking at themselves on their screen, or looking at the interviewer - but not at the camera! Please keep this in mind. You can either position the window showing you the camera images below your webcam, so you can look at them, or just decide to focus on the webcam when you are speaking or listening. It may feel odd and you might miss some facial clues, but the impression will be better than fixing your gaze on the screen (vs the webcam).
Agree with the points others have made - angle and lighting are key as well - you may also want to consider a professional background picture in case your actual background is not ideal.