Sorry for English but you stopped sharing your work on HC so I need to write it here
I use HD with H3 Complete and marked "No CD".
IMO, bugs about which you write above appeard with version 2.58 and "Fix gameplay Bugs" option.
If you turn it off, HD is more stable.
i.e.
If you run (new/load) game in MP HotSeat mode with "Fix gameplay Bugs = 1" and put the mouse cursor under any "Shrine of Magic", game always crashes. With "Fix gameplay Bugs = 0" it is OK.