How can I stop my iPad from automatically removing or hiding apps?

You appear to be using the App Library to open Apps that are installed on your iPad - the App Library icon (a shortcut to the Library) appearing at the extreme right of the iPad's Dock. Apps within the App Library are automatically grouped; these automatic groupings are determined by metadata - and cannot be edited.

Depending upon your iPad's settings, Apps that you download and install on your iPad may only appear within the App Library instead of your iPad's Home Screen(s). This behaviour can be changed in settings:

Settings > Home Screen & App Library > [Newly Downloaded Apps] - set as preferred

Apps already downloaded, that don't have a corresponding Home Screen icon for quick access, can simply be added from the App Library.

The iPad User Guide is a rich source of information and guidance for iPad and iPadOS. Here is direct link to the relevant section of the Guide that describes how to access and manage Apps within the App Library - including how to install Apps on you Home Screens:

Find and use your apps in App Library on iPad - Apple Support

This is not normal iPad behavior

You might want to ungroup your apps

Go to the Home Screen. Touch and hold the Home Screen background until the apps begin to jiggle. Tap the folder to open it, then drag all the apps out of it onto the Home Screen. When the folder is empty, it's deleted automatically.

And after that, restart the ipad to see if the issue persists