NLN
I am not condoning the use of ROC and have never used it myself, I am merely just stating my view through experience and what I would do. I admit there is room for error by using ROC and not sure whether the use of it has ever been included or banned in any official ATC documents. As I said I think it is a fall back from many years ago.
In answer to your questions:-
On initial contact to an approach unit what does an a/c have to say? When calling on box 2 what does the a/c say? Is it the same?
If the a/c took the wrong call to QSY then I would ring the unit it was being transferred to immedietly to advise.
If I didn't have any radar to see the a/c then I wouldn't expect a ROC, nor would I expect it to be published instructions.
Out of experience I would always announce that I was calling on box 2 to avoid confusion and also tell the controller that we are still with the controlling sector.