BARKING
As a dog owner, I have to say that all dogs bark. They might not be suffering from separation anxiety/boredom etc but all dogs bark. Its instinct. Mostly in response to another dog's bark or noise. Since both your immediate neighbour and fiance have already said that the dog doesnt, I suggest you request help from your neighbour in writing a letter to vouch for you, if you havent already.
Go on the offensive. Send the letter as well as proof of travel from Switzerland to the agency disputing the complaint. Make sure that you explain clearly to the rental agency and convince them re the barking.
LAUNDRY ROOM/COMMUNAL AREA RUG
What Frau Melloncollie said really is spot on when she brought up "Hundehalter haben viele Pflicht, kein Recht" . We have to understand that not everyone is fond of our 4 legged friends. Unlike screaming children and babies, people are less tolerant towards dogs. Hence, as dog owners, we need to be much more proactive in ensuring that our dogs arent causing a nuisance with their barking/hair because trust me, as you've experienced for yourself, the anti-dog squad will definitely make sure they'll pick on anything and everything whenever possible.
I wash my dog's bedding every other alternate week. He has lots of hair and I make very sure that I clean out the dryer extensively. My new-ish neighbour saw me in the laundry room and asked if i could stop washing my dog's stuff in the machine because its "unhygenic". I asked him innocently if its a better idea if I should wash it in the lake across the road instead, and that he should join me washing his smelly running shoes there instead of dumping them into the machine. And for the record: no, he isnt Swiss.
Also with regards to the communal areas, yes you pay for NBK and the likes for communal cleaning, but it does help if you also make sure that you go the extra mile and clean those areas if need be. If you know that someone's anti-dog in your building, muddy pawprints on the stairs are definitely going to make them pop a gasket Have a towel in the hallway pre walk and try to clean the dog as much as you can after wet/snowy walks. You can only do your best to ease the situation and not give them ammunition to complain.
Finally, I am going to suggest joining the mieterverband to know your rights as a tenant and the dog. If you can afford it, move to a much more dog-friendly building. No use stressing over this unnecessary pettiness and harassment.
Good luck and I hope everything works out for you.