What kind of vent is pictured here and what are its uses?
What you show in the picture is known as a One-way Breather Vent (there are also two-way).  Its intended purpose is to provide pressure equalization throughout the roof system and also provides ventilation for the insulation system.

In some buildings that do not have a vapor barrier (even some that do), pressure changes inside the building can be forced upwards through the plywood sheeting and potentially cause the roof system to pillow upwards and damage its adhesion. The breather vent gives the pressure an avenue to escape without causing any damage to the roof.

I would not recommend venting anything through the breather vent.  Most of the time breather vents are install through the roof system and do not penetrate through the substrates.  By cutting through the insulation and plywood you're essentially turning into a general building/attic vent and losing some of its effectiveness.