There is no paste-in-place yet.
Okay so in short, the issues appear to be:
Can't copy-paste groups between levels.
Confirmed. It only happens between different instances of the editor. Or in other words: If you copy a group, then create / open a new level in the same editor window, you can paste the group. If you try to paste it in a different editor window, it usually won't work. (There are cases in which it will, but I would suggest not trying to rely on these.)
I know exactly why this is occuring, but it will be a tricky one to fix.
Erase flag on group pieces
To expand on this (mostly as more-specific notes for my own future reference): This is specifically in reference to that if an instance of a group piece is marked as "Erase", the "Erase" flag is cleared when loading (there is no problem when saving). It consistently happens with instances directly in the level; it does not always seem to happen with pieces in groups. At this stage I don't have the faintest clue why this could be happening - the loading code shouldn't even treat the two differently aside from loading the group piece if needed. Most likely a bug relating to doing / not doing that.
EDIT: Confirmed, it was prety much as suspected. Very easy fix.
Pickup skills reverting to unassigned
The skill is saving fine, this is again an issue with loading. Most likely an oversight in the new level loading code. EDIT: Confirmed. The loading code does not even attempt to load the skill, let alone do so correctly. I still need to fix this one, I've just determined the cause so far.
In general, it should be assumed that any bug that could be related to loading, probably is, until that's ruled out - because the code to load levels has been entirely replaced in V1.25. (This in turn was due to needing a better implementation of parsing NeoLemmix text files, in order to properly support the grouping feature. Only level loading uses the new parser; objects etc still use the old one, so similar bugs are not a concern there at this point in time.)