As a native speaker, I have to say that I would never use the second sentence, "Did you see what they had done to our city?", except to describe a state of the city that has passed. For instance, if someone had painted a wall, I would ask a friend, "Did you see what they have done to our wall?". If that paint was then washed off, the question would be "Did you see what they had done to our wall?" as that state of "painted" has passed.
As for the difference between sentences three and four, I do not see any. It's just as correct to say "Did you see what they have done to our city?" as it is to say "Have you seen what they have done to our city?". However, I would say that if you asked somebody returning from the city, you would say sentence three rather than sentence four, as "Did you see" implies that they could have seen as opposed to "Have you seen" which implies that they may not have had the chance yet.
For instance, if somebody went to a cinema the same day that a new film was showing, I'd ask "Did you see that film?". If we were talking about the film, and I didn't know that they'd been to a cinema, I'd ask "Have you seen that film?".
As for the fifth sentence, you're absolutely on point.
This is the difference between the simple present tense, "Why do you take it?" and the present progressive tense, "Why are you taking it?".
The simple present tense is generally used for recurring actions. The present progressive is generally used for actions occurring right now.
So if you say "Why do you take it?" you are asking about a regular action, which probably happened before. If you say "Why are you taking it?" you are talking about only this current instance of taking.
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