Welcome to the forum, you'll find the members on here are a friendly and helpful bunch :)
In my experience the windows on both doors should go down slight when you open the door. However it isn't unheard of the components inside the window mechanism to wear out. However I have also found that if the battery is low on charge or has been completely flat then this can also cause this issue.
I'm unsure on the carbon build up though. I can't say I've come across it on here before on the 2.0 petrol TT.
Cheers :)
Steve