"We went into a mosque."
Translation:Wir gingen in eine Moschee.
In everyday German many people often use present perfect instead of preterite, yes, but if this lesson is about past tense, then preterite should be at least the preferred translation.
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I assume you are talking about the case after two-way prepositions (here: "in"), because other prepositions have a fixed case tied to them.
After a two-way preposition you use the accusative when referring to a motion. This has nothing to do with whether this is in past, present or future.
Does it necessarily mean that I didn't go inside? Or is it just more ambiguous than "in eine Moschee"? I am only asking this because I recently read an article about German prepositions and it says that "zu" could be used instead of "in", "auf" and "an":
"In general, you can replace the proposition in with zu, especially if you don’t want to stress the fact that you’re moving into a house, a building, a car, or anything else, when it doesn’t matter."
It does not necessarily exclude that you might go in, so you usually say "zur Kirche" or "zur Schule" (with a definite article!), if you emphasize the institution. If you talk about the building, you'd rather use "in". "zu" with an undefinite article would strongly suggest that you go only to the building, but not into it.
I have read the article, and the given examples are correct. But the "rules" are nevertheless too simplicistic, because all the mentioned examples fall into the "institution" category using a definite article. Cases like the given sentence here are not covered at all.