Auto arrange icons on iPad Air 3

Unfortunately no. You need to move your icons to fill the first page.

However, that should only happen with the first page, and new downloads by default appear in the second page or subsequent ones that have space for them and continue appearing there until there is no more space of them. At which point a new screen is created.

You can use the "Reset Home Screen Layout" in Settings->General->Reset option, and it will place all built in Apps back in their original locations, and all downloaded apps will be organized alphabetically, and take up all available space on all screens as required.

See here: Move and organize apps on iPad - Apple Support

There's no way to have gaps between icons other than the margins between them.

If you move an app in a page, the apps automatically rearrange to fill in the space left by the App.

If you have less than 20 Apps or folders in a page, then you will have space left over in that particular home screen.

If you are referring to the margins between Apps, that cannot be filled. The screen is a grid that has pre-defined margins for the icons.

Automatically, no. You can physically and easily move apps from one screen to another screen by resting your finger on any app until they all jiggle, then with your finger move the apps wherever you want them to be.