I have spent the last 5 1/2 years in Köln, and I love it here. I rarely go up to Düsseldorf, not because I hate the place, but because Köln has everything I need. As an Aussie they have an Aussie shop, the town is located near both D-dorf and Bonn - midway point if you like; the airport is easily accessible via public transport, and the city center impresses me more. The access to the A4 autobahn makes getting to Belgium or Holland (or the UK for that matter) an easy drive.
I find Nordrhein Westfahlen in general to be a great state with a great attitude. (My personal preference, though is Niedersachsen, especially on the coast in Freisland, but that's just me). If you have a car, it really doesn't matter where you live. D-dorf is 30 min north, Bonn is 30 min south, Belgium/Holland 45 minutes west from Köln. Frankfurt Airport is connected by ICE trains in 55 minutes from Köln hbf.
My preference for beer is Jever pilsner. Kölsch and alt are equally as shit as each other if you ask me. I also tend to buy very strong Ukrainian beer from a Ukraine supermarket, and Polish vodka from the Polish butcher across from the Ukrainian in the Chorweiler shopping centre 13 km north of Köln.
Both cities have the Rhein, Düsseldorf being the capital of the state. It is more modern, but has less flare. When I went for my german speaking test prior to my official settlement papers, I was greeted as a local. That was one of the nicest feelings. Despite hearing horror stories of German attitudes to foreigners, I have experienced helpful and friendly staff at the ausländeramt. To the point that they were sad to no longer see me once my officialization had been finalised.
Unfortunately no, the GEZ is paid directly to them seperate from cable fees. Similar to the UK, they have people who check whether you are watching illegally. You are required to pay GEZ regardless of cable, car-radio, TV, internet (the internet is new, and the reasoning is you can listen to the radio via internet).
You should be able to head down to the Ausländeramt and request the ammended sticker in your passport. It is your right to have it, and the only thing causing potential delays is the fact that even the Beamter themselves dont quite know about it as it is relatively obscure. In my case the evidence of Renteversicherung, Tax certificates, and the police certificate were all utilised from my normal NE application. I simply told them my work may be required to work in other EU lands.
Stick to your guns, because you are entitled to the Daueraufenthalt EG without any problems.