What is the difference between “flat” and “apartment”?

Does "purchase order" mean the the paper she's looking at? Does "order" itself have a meaning of "printed form"?

The document Rachel is looking at is actually a "purchase order form," but she has shortened it to "purchase order" when speaking. "Order" itself does not mean "printed form."

What does “let someone down” mean?

To "let someone down" means to disappoint someone by failing to do what you agreed to do or were expected to do. Ex: I'm relying on your help tomorrow, so please don't let me down. Ex: I don't want to let my teachers down in the exam.

Isn't it possible to put "them" after "picked out"? Like "I've had picked out them".

No, it would sound awkward to put "them" after "picked out". Although "them" is the subject of this sentence, it would not be correct to place it after the verb in this sentence because it sounds awkward. The subject normally goes before the expression "to pick out". If you don't include the verb "had" in this sentence, you could also place the subject in between "pick" and "out". Ex: Do you like my shoes? My cousin picked them out for me. Ex: She already has name ideas for the baby. She's had them picked out for several years.

What's the main difference between a "part" and a "piece?" Or are they interchangeable?

Good question! "Part" and "piece" have a similar meaning but cannot be used interchangeably in all cases. "Piece" is a single, individual component of something else. "Part" (noun) is a section or amount of something else, which when combined with other "parts" forms the whole thing. Pieces are always parts, but parts are not always pieces. Ex: The engine is one of the most important parts of a car. Ex: A car engine is made from hundreds of pieces.

What is "fashion week"? Is it some special week about fashion?

"Fashion week" is an annual event in the fashion industry where brands and designers display their newest collections with models on the runway. It lasts about a week each year.