could you use "olika" here instead of "en annan", would both be correct to say "different"?
No, "olika" is a comparison between two or more things, but "avdelning" is in singular. So if you have two departments, you could say "hon arbetar på två olika avdelningar" (she works in two different departments), but it doesn't transfer into singular so you can't say e.g. "hon arbetar på en olik avdelning".
I can't hear even a weak 'v', in this. In fact, I think I hear aldelning. I'm not convinced that robot's lower lip and upper teeth ever meet :) . In Forvo, I clearly hear it with 'v': http://forvo.com/word/avdelning/#sv
Do differences like this represent how different Swedish speakers have different speech accents, or is this a shortcoming of the current TTS ? Is it just this specific word that likes to have 'v' so weakened and Duo knows this? Also, I guess I'm hearing the stress on first syllable in Duo, but on the second syllable elsewhere.