Luxembourg... or Liechtenstein?

Luxembourg has an airport and Liechtenstein does not! It might pay to say "Has German as an official language" because Luxembourg actually has three, one of which is German. Since I've visited Vaduz, and it's on the edge of the Rhine River, it's hard to work out if that counts as flowing "through" it.

Vaduz is definitely by/on a river. The Rhine is just next to the city. Just look here

Vaduz is on river Rhine. Fix that.

Yes, the river question is dubious. The Rhine river is right next to Vaduz, and even though you want to not count it, there is the Giessen river that flows right through the city.

Also, question #14, "double landlocked" means "landlocked by two landlocked countries", not just two countries. Uzbekistan is a double landlocked country, because all its neighbours are landlocked, but it has 5 of them, not just 2.

If you want to be extra nitpicky, you could argue that Liechtenstein is named after a castle as well. The info you probably based the answer on is that the country was named after their royal family. That family, however, named themselves after a castle in Lower Austria. It's probably up to everyone's own interpretation if that counts...

14. Is double landlocked (landlocked by two countries)

double landlocked is landlocked by countries that are landlocked too