In my school, the playground is too small to have a 400 m track, so the school constructed a 200 m track instead. Does this mean that we need a smaller stagger for the race tracks in my school (i.e., smaller than the stagger used in the Olympics), for the same 4 x 100 m relay race?                                                                    [Page No. 118]

No. The stagger between two adjacent lanes does not depend on how big the track is; it depends only on the width of a lane. The extra distance an outer-lane runner covers comes entirely from the curved ends.
Over one full lap the two bends together make one complete circle, so the extra distance for the next lane out is $2\pi(r + w) - 2\pi r = 2\pi w$, where $w$ is the lane width.
This has no $r$ in it, so a smaller track gives the same per-lap stagger as a large one, provided the lanes are equally wide.

There is a twist, though. A $4 \times 100$ m relay is 400 m of running. On a 400 m track that is one lap, but on a 200 m track it is two laps. Going round the bends twice means the outer runner falls behind twice as much, so the total stagger the race needs is about double i.e., larger, not smaller. So, the small track does not call for a smaller stagger at all.