Translation:The young woman was thinking about her mother.
Doesn't that sentence imply that she was thinking about someone else's mother? As opposed to «Девушка думала о своей матери»?
It depends on the context. Yes in some cases this sentence may mean that she was thinking about someone else's mother.
In Russia woman can be young or old, doesn't matter. Without young is correct too
Why devushka HAS to be a girl/young woman? I know it differs from zhenshina, but is it so distinctly different that it's marked as a mistake?
It is the prepositional, as called for by the preposition о/об/обо. In this case it is the same word as in the dative, though!
Thanks. I tried to wise up, but I don't think I was successful. There is no real rule to decide which case comes after a preposition, is there? The same preposition depending on context (mostly the verb) may govern different cases?
It depends entirely on the preposition and the meaning you're trying to convey, yes. Some prepositions can call for different cases depending on context but the prepositions о and при are always associated with prepositional.