What is the interstage used for, how big it is and is it deorbiting separately?
On the Falcon 9, the interstage is attached to the top of the first stage and remains attached to the first stage.
It also contains the mechanisms for the grid fins, and the grid fins themselves.
The nozzle of the second stage sticks out from the bottom of the second stage, and is covered by the interstage.
The interstage provides a mount point for the second stage to attach.
The Falcon 9 interstage, which connects the upper and lower stage, is a carbon-fiber aluminium-core composite structure. Reusable separation collets and a pneumatic pusher system separates the stages. The original design stage separation system had twelve attachment points, which was reduced to just three in the v1.1 launcher.
On the Saturn V the first stage interstage was a ring that was discarded and reentered on its own as seen is this nice question. The second to third stage interstage remained attached.
On some boosters (looking at you Soyuz), where the next stage actually starts firing before separation, the interstage will be a mesh/grid that allows the startup exhaust out during second stage startup.
To accommodate the third stage, the core booster is crowned with a lattice structure, featuring 12 connectors and six locks, which are designed to be cut with pyrotechnics to separate the second and third stage in flight.
The lattice structure makes it possible to ignite the engine of the upper stage moments before the separation of the core stage below it. A special cone-shaped, titanium-covered deflector above the instrument section is designed to facilitate the flow of hot exhaust gas from the third-stage engine, when it begins firing.