Does stabilization make the image quality little worse?
Depends on the camera, the stabilizing technology and the amount of work the "stabilizer" has to do to stabilize the image. A tripod is a stabilizer. When used on a steady surface, a tripod is a very common and useful way to stabilize video during video capture when the camcorder is mounted to the tripod. I have never seen where any of my tripods make the image quality worse. A Steadicam articulated/counterbalanced arm connected to a vest is a stabilizer. When I jump, run or walk using my Steadycam Solo with a Steadycam vest/arm stabilization system, the captured video is great. A camera crane is a stabilizer system. I use a Kessler camera crane on a Hercules head on a K-Pod tripod... and the captured video is rock solid. A new kind of handheld stabilizer for smaller cams can work well if the shaking is not too bad. These little stabilizers can be motorized to help keep the camera view level. When they are used in a way that includes too much shake or motion, they can cause issues - same with built-in... ... Optical or electronic image stabilization. Humans were not built to be steady. If there is a little shake or vibration, then the built-in anti-shake stabilization system can help mitigate that, but any more than a little and expect the image quality to degrade - the most common "artifact" is the video gets "wavy" when the motion or vibration is more than the built-in stabilizer can handle.
Depends of the type of stabilization. If it's digital stabilization, then yes, image quality suffers due to the way that the image must be zoomed-in on, which reduces resolution. If the stabilization is optical, then the image isn't affected. If the stabilization is created by a gimbal or tripod, then there is no effect on image quality.
If it is optical stabilization, it should be better quality. Digital stabilization may reduce resolution slightly to get less motion blur.