Altium PCB Editor Room "stealing" tracks from other rooms
A really dumb workaround which I found uses what is probably another bug: The fact that if you select an object or a track and then a room only that object and the room polygon will be moved.

So by selecting the components and tracks of the room first. And then selecting the room, all components and tracks of that room can be moved without disturbing any other traces. Disclaimer: I have no idea if this method has any negative consequences.

This is the easiest method that I have found for achieving that selection combination for a room which is over other tracks and components on other layers:

Step 1: Select Components:
Done once: To prepare we need to be able to cross probe multiple components at once: Enable Prefrences->System->Navigation: Check cross Selection box.

Go to the schematic instance of the room you want to move and hit Ctrl+A in that sheet instance. This selects the components of the room in the PCB editor since we enabled the auto crossprobe function.

Step 2: Select the tracks:
Go to the PCB editor and right-click on the room that should be moved WHILE holding shift: 

RMB{in room while holding shift}->Room Actions->Select Room Connections

Now you can release shift, both the components and the tracks should now be selected.

Step 3: Selecting the room:
 Shift LMB click on the room to select it. Now the room, the components, and the tracks are all selected. 

Step 4: Move the room: 
You can now move the room and its tracks without moving any other tracks.

Stupid? Yes.