Ok, I just want to make sure I'm correct in what I've discovered.
1. You cannot paste a layer between two compositions in After Effects.
2. Once you create a composition with a quick time movie then place that composition in another composition you cannot move that composition in the timeline and you cannot resize the movie.
...ok I lied I found I could at least resize, but still can't move in timeline unless I enable time-remapping.
Why does this layer require time remapping but the others don't. They are even more complex.
I tried to replicate your issue and I had no problems moving the precomp in the main comp on the timeline. From your stills it looks like you are stretching the precomp - is that what you are referring to when you say moving?
I did stretch it, but not that I care about that. I also have a mask on the video. I don't think that matters though. It seems like, the embedded composition settings (0:00-1.17 sec) are also the main composition settings when it's embedded and I can't change them unless I check time remapping. I also previously used layer monkey to add lyrics (to main) then I removed them. I wonder if it changed some setting in the main comp.