swing by my house/would you mind

[Saneta]

http://www.english-test.net/toeic/listening/could_we_swing_by_my_house_on_the_way.html

Could we swing by my house on the way?=
Could we pass by my house on the way?=Could we go into my house on the way for a while?

Excuse me, would you mind taking our picture?
No, not at all. -- CORRECT
Yes, I would mind.- WHY NOT?
Sorry, I'm in a hurry-WHY NOT?

Many Thanks!!!

[Beeesneees]

swing by = 'go via'. The speaker wants to go to his house (possibly to pick something up) on the way to the final destination.

As for the other question, although a question frmaed 'would you mind...' is a polite way of saying 'please will you...' I wouldn't expect the negative answer to be 'yes, I would mind'. :
More likely is:
Would you mind taking our picture?
I'm sorry, but I can't do that

However, I see nothing wrong with, "I'm sorry, I'm in a hurry."
The question is faulty and should be changed.
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Cheers m' dears!