Learn English – Shall I write, I am going to buy or I have bought

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.

Well, if you look in a dictionary for the words time and day you will certainly see the difference.

As for using any day instead of any time, well, yes, it's possible:

• You can come at any time. (or You can come anytime)
• You can come (on) any day. (the preposition is optional and I always omit it)

Both are correct.

The first sentence you wrote is really awkward, I would rephrase it using ever-

• Have you ever seen this before?