I assume they did the same as they do at the Super Bowl. The music is tracked along with an emergency vocal track. Celine was obviously singing live. There was a pianist earlier in the show playing a Ravel piece that was obviously tracked.
Yes! And also big, good knobs on a large panel.
Well, in this case I'm almost sure there were problems with the video camera, because I noticed the issue while editing, much before uploading the video on YT. While I was shooting the video, I had set up the lights and the camera settings exactly like in previous videos, but the outcome was so bad that for the first time ever, I had to attempt corrections in the editing stage. It was only partially useful, and it added some artifacts to the image (you can notice that on my hands, for example).
Btw, thanks for the suggestion, I'll give it a try.