How Well Do You Know Your German Two-Way Prepositions?

Translation: Karl, please put the vase on the table.

The "an den" choice was incorrect because a table is a horizontal surface and "an" is used for vertical surfaces. The "auf dem" answer was incorrect because this is an accusative (wohin?) situation, not dative. The vase is moving towards the table. 

Translation: Today you don't have to stop at the border.

The accusative phrase "an die" doesn't work here because it would indicate motion towards the border. But we're stopped AT the border: "an der Grenze" (wo? - dative). A border is treated like a vertical barrier, so "auf" doesn't work here either. 

Translation: My glasses fell into the water!)

English "into" is almost always "ins" (in das) with "das" nouns. Wohin? - where to? Into the water. 

Translation: Today many people were at/in [the] church.

This question answers the question where (wo?) as in a location, so it takes the dative case. 

Translation: Two years ago we were in Switzerland.

Dative plural nouns usually end in "n". "Vor" with time is dative. 

 Translation: From Venice we flew over the Alps to Munich.

The Alps are "die Alpen" - here "über die Alpen" means "across the Alps". If we were hovering over the Alps (like a cloud, wo?), that would be "über den Alpen".

Translation: I want to hang the picture behind the desk.

"Hinter" is the opposite of "vor" and behaves just like that preposition. These kinds of phrases change to dative if an "auf" prefix is added to the verb: "Ich will das Bild hinter dem Tisch aufhängen." 

Translation: Can't you read between the lines?
This question asks where (wo?) not where to, so it takes the dative case. 

Translation: Hans was in the USA for two months.
Wo war Hans? - dative plural answer; in German the U.S. is plural ("the United States [pl.] are..."). 

Translation: The book is lying [over] there on the table.)

Horizontal surface (auf) in the dative (wo?) 

Translation: I'll wait for you in front of the movie theater/cinema.)

Where (wo?) will you wait? = dative. (das Kino > dem Kino)

Translation: We're spending the night under the open sky.

Wo? - dative (of 'der Himmel') The adjective "frei" takes a strong -em ending because there is no article in front of it. 

Translation: I'm sitting next to my wife.

Wo? - dative (of 'meine Frau') 

Translation: Maria, you have a long day ahead of you.

Dative plural nouns usually end in "n". "Vor" with time is dative.

Translation: Are your children already going to school?

Where are they going to (Wohin?) Motion requires the accusative case. 

Translation: The rider mounts [climbs on] the horse.

Wohin? - where to? - "onto the horse" - auf das 

 Translation: The dove/pigeon is sitting on the roof.

Wo (where?) is the dove sitting? (Dative case).

Translation: She sat down at a table in the corner.

While "sich setzen" is accusative (wohin?), the phrase "in der Ecke" is dative because it refers to where (wo?) the table is located. 

Translation: The teacher writes the vocabulary on[to] the blackboard.

In German, when you write on the blackboard/chalkboard, you write "onto" it. Once the writing is there, it's "an der Tafel"! (Wo? - vertical surface)