Without seeing your comp window with the changed or animated properties of your problematic layers revealed I can't give you specific instructions. The first thing to to would be to find all of the keyframes, and keyframes in any pre-comps, and then adjust the keyframes and the out points of the layers. It's just like editing any other kind of video, if you want to change the length of a segment you need to change the in and out points.
As for the problem of locking new elements to the moving background the original elements were either pre-composed with the background or tied to background animation by parenting or expressions. Again, without seeing your timeline with all properties revealed I have no clue how this was done and since there are about a hundred ways I can think of that will duplicate their sample video I can't give you specifics without more info.
To reveal all the properties in your timeline so we can see what's going on make sure the parenting column is visible, select all layers and press the u key twice. Then take a screenshot and post the full resolution screenshot on this forum. If you cant "insert image" the first time you try, close the pop-up and try again because sometimes the java running the feature doesn't work the first time.
Rick, thank you. I am completing a render proof, and then will re-open AE and do as you asked. I appreciate your help.