The time code in the Caption Panel is the timecode within the caption stream. If the first caption starts at zero and there are no disruptions of the stream in the timeline, the timecode will match. Otherwise, it will not. Note that it is the time of the first caption, not whether the stream shows as zero in the timeline. This is not an error, it is just the way it works.
I can't tell if your captions start at zero, but you have cut your stream, so the times won't match after that point. That is not necessarily a problem.
I have not been able to create the problem of the caption disappearing when adjusted in the timeline. But others have had big problems with this. I don't know of a foolproof workaround - other than not adjusting in the timeline.