Learn Danish: Hun besvimede i varmen. - She fainted in the heat.

Yes, but it changes the nuance slightly. I varmen translates to "in the heat," focusing on the environment she was in when it happened. If you want to explicitly state that the heat was the medical cause (fainting from heat), you would use af: Hun besvimede af varme. Notice that when using af for a cause like this, Danish usually drops the definite article.

Why does this sentence use the definite form varmen instead of the indefinite varme?

Using varmen (the heat) grounds the sentence in a specific context, referring to the hot weather or temperature of that particular day or room. If you were to say Hun besvimede i varme, it would sound unnatural for a single past event, sounding more like a general, scientific statement (e.g., "she generally faints in warm environments").

As a regular past tense verb ending in -ede, how is besvimede actually pronounced in natural speech?

While spelled with three syllables at the end, the -ede suffix is heavily swallowed in spoken Danish. The final -e is typically dropped, and the d becomes a very soft, almost unnoticeable sound (a soft d). In conversational speed, besvimede sounds closer to "be-SVI-með" or even just "be-SVI-me".

In English we often say someone "passed out". Is there a phrasal verb for this in Danish, or is besvimede the only option?