iPad OS update

Yes - the installed Operating System (iPadOS) occupies some of your device internal storage. You can see your internal storage from iPad settings:

Settings > General > iPad Storage

By example, space occupied by the iPad’s Operating System is colour-coded in orange - showing the proportion of internal space that it occupies:

When you install a system software update, the update is initially downloaded into available free-space - and here the compressed download is unpacked (requiring additional free space) and verified.

Assuming that the downloaded software is correctly verified, your iPad will restart and replace (or update) the existing version of iPadOS. After the process completes, the update package is automatically deleted (as it has done its job) and the iPad restarts as normal.

Yes.

Any OS installed on ANY computer or computing device takes up internal drive storage space.

The OS is not stored on a CPU chip.

It is loaded into the internal storage drive.