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.
If he's talking about something that has already happened, then the correct wording would be: If I did that.
If he's talking about something that could have happened in the past, then the correct phrasing would be: If I had done that.
Wikipedia has a good article on English conditional sentences. When expressing a hypothetical situation, "would," "had," and "were" are used instead of other verb forms, e.g.:
(Hypothetical)
I knew he would do poorly if I were there, so I didn't go.
(Not hypothetical)
I knew he would do poorly while I was there, which is why I went.
In spoken English, people often find this phrasing awkward. For example, this sentence is correct:
If he were there, things would have been different.
However, many people would change that phrase to the following:
If he was there, things would have been different.
Even though the second sentence is incorrect, it is more comfortable for many English speakers because the phrase "he was" is far more common than the conditional "he were." Also, "he were" is incorrect in every instance except a conditional sentence. Many English speakers do not know (or do not care) about this particular case of subject-verb agreement.
So, from the example you've given us, the "that" which your speaker is talking about ("if I did that") is probably the thing that he chose not to do.
Best Answer
In the U.S. we usually say; I went to school at St. Xaviers.
and for the second sentence we say: "I had taken my 10th Boards from there".
[We don't need to say school because from the context we know you're talking about school so it becomes repetitive.] Also I use 'taken' because I assume it was you who took the test not you who was giving the test.