It sounds like you want to actually move (not copy) your elements laterally over the entire length. If you can do that at all, I'll be impressed.
If your elements are derived by offsets from other linework (other linework that will remain where they are now), then you should be able to just change those offsets with the various inputs that we have access to. If they're located according to snaps, you'll need to establish new elements to snap to, I would guess.
Once all that is done, I expect that your element will be longer or shorter than it is now. In my own experience (which is limited, admittedly), if your profile is established by static lengths between VPIs, the profile (at least the stationing thereof) will not shift along with your plan element; the vertical aspect may shift, if you snapped to key points. If your VPIs are established by specific crossings, it may match.
In the end, I'm pretty sure it'll depend on your scenario.
To answer your last question, you'll probably find that a "from surface" profile should update with any change to the plan element. If you want this not to happen, you'll need to keep an element where your current element is, and then project from that element onto the offset element.