The warm moist air from the bathroom may be condensing in the cold vent pipe.
Snow melting on roof above bathroom.
Has the snow melted yet.
Another part of your dripping problem may be the way the vent pipe is routed.
It might be blocked.
If it has then the black flies have arrived.
Warmlyyours ice shield roof gutter deicing cable system provides a reliable solution for ice damming that occurs in gutters downspouts overhangs valleys and bottleneck areas between roof features such as dormers or adjacent rooflines.
The pipe exits the roof above the right side of the bathroom window as shown in the pics below.
Snow catchers at the base of metal roofs hold back sliding snow near the eaves.
Causing plaster around window to bubble and creating bubbles and cracks in the paint in the ceiling below.
Snow disappears quickly from roofs above attics with little or no insulation but will stay a long time when.
Venting through the soffit is probably your best option.
Ice and snow accumulating on your roof or inside your gutters can cause costly water damage to your home or business.
How the snow melts on your home s roof provides clues to attic insulation needs.
Paint the out side to prevent water damage from the melting snow and make sure it fits snugly.
Water will run down the roof and can freeze at the eave which is cooled from the underside figure 5.
Sometimes when it s raining and even when it was below freezing snow outside we ve had water dripping inside a bedroom window.
It could take a while for the warm moisture in the vent to melt the snow above it.
That one just above the lower roof looks especially susceptible to it.
A couple of eye hooks on each side to hold the board in place or maybe some short pieces of 1x2 with a screw through the middle that you can rotate to horizontal.
However i suspect that your snowmelt problems may be deeper than just an omitted roof vent especially because you said that melting occurs even when the fan is not operating.
With the snow that roof is probably too dangerous to walk on so i m sure you did not look down the vent with a flashlight.
Exposed shingles or metal can easily be heated to 70 f 40 c above ambient air temperature and cause melting of the snow immediately next to the exposed roofing.
It might not be.
Definitely coming in from above as we can see the water dripping from the top of the.
Roof vents are also subject to blockage in heavy snow and they have been known to drip condensation back into bathrooms.
It s a slight dormer window in a sloped mansard second story roof.