1. 1週間くらい休みがとれそうですから、家族と一緒にハワイに行こうと思います。
My transation: "It seems I can take a week's holiday, so I think I'll go with my family to Hawaii."
Is this とれそう from the potential form of 取る ("take"), per my translation "seems I can take"? There are so many different versions of とる and とれる in the dictionary, it's bewildering...
Yes.
とれそう=とることができそう
2. 異国情緒あふれる街並みが続く。
Translation given: "The streets are filled with an air of exoticism."
I don't get the grammar of this sentence. Is there a missing particle or something? Should it be 異国情緒があふれる modifying 街並み? That would kind of make sense to me.
異国情緒あふれる is a set phrase, meaning "full of exoticism", "very exotic".
3. すみませんが、勉強中なのでもう少し静かにしてもらえませんか。
My translation: "Excuse me, I'm in the middle of studying, so could you possibly be a little quieter?"
Here I am unsure about にして. Should I think of 静かに as "quietly" and して as "acting", so literally it would be something like "could I receive the favour of your acting a little more quietly"?
静かにする means to be(come) quiet/make (themselves) quiet.
静かになる to become quiet (spontaneously, without particular effort)
静かにする to become quiet (intentionally, as a result of effort, etc.)
4. あしたお返ししますので、この本を1日貸していただけませんか。
My translation: "I'll return it tomorrow, so could you lend me this book for one day?"
Is this OK? (In fact, in English, the sentence is slightly odd because "so" does not quite fit the relationship between the clauses. I'm not fussed about this provided my understanding of the Japanese is essentially correct.)
Yes, your understanding is correct.
Its connotation is something like "I'll return it as soon as tomorrow, so could you lend me this book just for one day?".
"Could you lend me this book just for one day? I'll return it tomorrow."
5. 大学の経営が厳しいらしい。それは若者の人口が減っているからだ。
I translated this as "The university management seems strict. This is because the numbers of young people are decreasing." However, I don't really understand my translation. What has one thing got to do with the other? What am I missing?
Here 厳しい means "difficult".
The logic is:
To run a university with healthy financial conditions, they need a lot of students. Decreasing of young population means they have less chance to get students who will pay the tuition which eventually contributes to the finance.
edit: Oh, Toritoribe san beat me!