It's most likely that the flap actuator(s) are knackered to be honest. It's probably worth getting the fault codes read to be sure. The actuators are expensive to repair IIRC because it means stripping the dash down.
I would imagine so yeah. AFAIK it's a case of taking the dash apart to get to the air vent flaps but it'll need some diagnostics to check if it's the actuator itself or not. Never done or seen it to be honest but the first step should definitely be getting the code read.
In vagcom aka vcds, on an 8P anyway but check yours aswell, there's a basic setting in hvac module 08 that actually tests the unit & sets it up as if were new, think was 001 with vagcom, give it a go & see what you get back as if its a flap then should show you which one like V153 etc, but staz is right, the main unit is housed behind the center console & it spreads across to the passenger side behind the glovebox, the main center unit usually houses the motors etc for flaps, although there's 1-3 actually on the part behind the glovebox, but its tad tight in there to get to & I unfortunately have a similar job to upgrade my climate at some point, not fun at all especially in winter.
Let us know what vcds reports as it might just need recalibrating to resolve.