A very warm welcome to Beoworld..
As you are talking about Freeview and Sky HD box, I am assuming that you have the integrated DVB-T/S module fitted to your Mark III Beovision 7, please confirm ?
If you have a DVB-T/S module for freeview and you have done a number of scans without getting the apparent missing channels, then the problem could be your aerial. What channels are missing from your Freeview listing ?
With regards to the Sky HD box apparently turning off the picture and sound yet still playing, it could be that the STB controller IR transmitter is not directly the receiver correctly on the STB which would certainly give you this type of symptom particular as you said you had the same problem with your old Beovision.