Don't know about the Audi, but the Tiguan is built at THE SAME FACTORY as the Kodiaq in Czech Republic, has the same platform and most of the same engine choices, suspension and brake caliphers so they should fit 99% (SHOULD IS NOT THE SAME AS WILL !!!!!!).
The Audi Q5 has a MLB rather than an MQB platform and different brake caliphers so they probably won't fit. The Q3 could fit but not that sure, due to the smaller size of the car.
I personaly never buy tyres at the car dealers, I go to buy them and rims at a specialised tyre/rim shop. Been doing that for more than 10 years and always get good prices and quality tyres. They will know what tyre size you can fit. For the Kodiaq it depends on the rim size, you can get the 235/55 R18 or 235/65 R17 for 18" or 17" respectfuly. You can get them wider (245 but probably not 255) or thinner 225. It doesn't matter what car they came for as long as they are the correct size.
For my SUV i buy Bridgestone Blizzak 255/60/R16 Studless for my Jeep and will for my Kodiaq (though different size) and go through 50cm of snow on a 60% grade incline, for summer i used Michelin tyres which wore out fast but were the best.
My advice, you have a new car, buy proper NEW winter tyres from a tyre shop and for summer use, if you must, second hand ones. In winter conditions GOOD TYRES are MORE IMPORTANT than a 4x4 drivetrain.