Why would you bother to get Audi to do the work? I could understand if the car was newish.
Buy yourself a used door off eBay and fit it yourself, less than £100 all done, pocket the rest to make up for the hassle.
I would say so, there's nothing to suggest the engine is seized. Your mechanic is trying to rotate the engine with the injectors still in, it could simply be compression that's stopping it from turning. These V6 engines are a big lump.
Its very common, in any car. Its usually condensation built up in the exhaust which turns into steam as the car warms up. It will usually disappear after a while when its all evaporated.
I'd get a 2nd opinion on the battery & alternator before doing anything. A faulty alternator certainly can affect the cluster, but also it could be the cluster itself.
I personally wouldn't rely on anything Halfords say. I'm not saying they are wrong but would trust an auto electrician more.