Word Choice – ‘Out of’ vs ‘Outside of’ City Compared

I have to admit I haven't consulted any sources, but here is what I'd say is "correct":

• On Saturdays, her sister Ann usually comes to stay with Mary for the weekend.

and

• What are you going to do for the weekend?
• What are you going to do over the weekend?

and

• We are going to Paris for the weekend.
• We are going to Paris over the weekend.

"Weekend" would not normally be pluralized when preceeded by the word "the". For example:

• Are you going to stay here on weekends?

And finally, to address "at the weekend": this is often seen in British publications, but I've never seen the phrase "at the weekend" in American English.

I think the preposition is necessary. The analogy with case in point is not quite apt as follows: "In which city" can be replaced with "Where", hence "Where are you located?" is correct but "In where are you located?" and "Where are you located in?" aren't. Similarly, "What time is the meeting?" or "When is the meeting?" is correct, but not "At when is the meeting?"