The word "Mode" is pronounced strangely here.
I know - I couldn't tell if she was saying "fashion", "murder" or "moon".
Sounds similar to this native speaker's pronunciation, to me: http://www.forvo.com/word/mode/#sv
It almost sounds like they're saying moden or modet.
There's a clear 'n' in 'mode' :(
Can't läsa also mean 'to study'? Or is it then used with a different preposition?
It can mean that, yes. But with the preposition "om" ("about") it's apparent she reads about fashion here rather than studying it.
So if it was "min syster läser mode" it means she's studying fashion? Or do you need another preposition?
It´s correct! You don´t need any preposition.
Min syster läser mode. - My sister studies fashion.
Jag läser svenska på universitetet. - I study Swedish at university.