"De antager, at pigen er perfekt."

Translation: They assume that the girl is perfect.

[dtgman]

Is the fast version or the slow version of "antager" in this sentence correct?

Version slow sounds more like "an tager" where fast version sounds like "an teger".

[miacomet]

The normal speech sounds right to me. The TTS can be a little temperamental. In this case, the a has more of an /æ/ sound, and the g is soft, so it has a /j/ sort of sound. You can hear a real person's pronunciation here: http://ordnet.dk/ddo/ordbog?query=antage

(Copied from your other question, but with a different link. :)

[DTU_UPCH]

Is the comma in the Danish version appropriate?

[GustavoCoe411014]

yes and no.

it is called the "grammatical comma", in which you put a comma between a main clause and a following subordinate clause.
but the "new comma" system, which is recommended, does not require a comma there before "at".

Both systems are accepted.