Shower – How to place a bathroom vent fan in a vaulted/cathedral ceiling

Yikes, this is really the first time I have ever heard of this configuration. Could this have been part of a previous central forced hot air system? It is not common to tie exhaust vents from a bathroom to any other living space. A dedicated exhaust fan vented to outdoors with a one way damper is a very important way to remove excess moisture from bathrooms and help prevent mold and mildew growth. It is not acceptable to vent a bathroom to another living space or into an attic. It is equally not right to tie this type of vent to any other air plenum. If you are feeling cold outside air through these vents, you are getting huge heat losses. There is no need to introduce outdoor air into your house intentionally unless it is done with specifically designed air exchange equipment in an extremely airtight designed house. I think you need to determine the routing of these "phantom" vents and come up with a plan to isolate the bath vents to the out of doors with a proper bathroom fan unit and block your heat form other areas from being wasted out an open plenum.

The vent doesn't need to be unclogged. You stated that paper is held to the register. That scenario shows that air flows through the vent unrestricted. The probable reason it might appear to be blocked (when the window is closed) is you reside in a well sealed building that has few drafts. With out going into detail about air flow; the reason you detect air movement when the window is opened is due to air flow from outside having a unrestricted passage through your vent. When the window is closed the air currents have to detour around the building walls.