Downgrade to iPadOS17

Try a forced-restart of your iPad - as this will often clear an isolated software issue:

Force restart iPad (models with Face ID or with Touch ID in the top button)

If your iPad doesn’t have the Home button, do the following:

1. Press and quickly release the volume button nearest to the top button.
2. Press and quickly release the volume button farthest from the top button.
3. Press and hold the top button.
4. When the Apple logo appears, release the top button.

If the problem persists after performing the restart, it may be necessary to reset your iPad settings. While this reset will not cause any loss of data, as a precaution, it is always wise to ensure that you have a current iCloud or iTunes backup before performing the reset:

How to back up your iPhone, iPad, and iPod touch - Apple Support

When you are happy to proceed, you can perform the reset from iPad settings:

Settings > General > Transfer or Reset iPad > Reset > Reset All Settings

If the problem still doesn’t clear, the next step is to reset the iPad to factory settings:

Restore your iPhone, iPad, or iPod to factory settings - Apple Support

Otherwise, you would be best advised to seek assistance directly from Apple Support. You can contact the Support Team using the Support link at top-right or bottom-left of this page. Alternatively and perhaps easier, you can initiate and manage your support cases using the excellent Apple Support App. If not already installed on your iPad, the App can be downloaded from the App Store:

https://apps.apple.com/app/apple-support/id1130498044

You cannot downgrade the version of iPadOS in your device. But iPadOS 18 still supports an external monitor as an extended display. Perhaps this will help:

Connect iPad to a display with a cable - Apple Support

Regards.

Stage Manager and Screen Mirroring are still very much present in iPadOS 18 - although my own testing suggests that iPadOS 18(.0) is perhaps a little more fussy about the connected monitor than iPadOS 17.x.

Consistent with iPadOS 17.x, your monitor’s screen resolution will impact the available settings and external screen capabilities. 

While undocumented, testing has demonstrated that 4:3 monitors with a display resolution of 1280x1024 and below are currently only usable with iPad screen mirroring; as such, associated settings to disable screen mirroring in favour of Stage Manager are not available in iPad Display settings.

When used with a 16:9 widescreen monitor, with a native display resolution of 1920x1080 or greater, the full set of Display settings - and access to Stage Manager - become available.

The iPad User Guide lists the upper resolution limit as 6K - but the Guide offers no lower limit, or suggestion of aspect-ratio, being a factor for compatibility:

Connect iPad to a display with a cable - Apple Support

As for rolling-back to an earlier version of iOS/iPadOS, this is not always possible...

All System Updates are digitally signed by Apple - and can only be installed whilst the digital signatures remain valid. After release of a new iOS/iPadOS version, the immediately preceding version continues to be digitally signed by Apple - however, usually within a few days, Apple will revoke the digital certificate with which the earlier version is signed.

However, at this time, iPadOS 17.7 remains signed and available. The process is not trivial; while it is currently possible to restore this version using a Mac or Windows PC, the process will fully reset the iPad losing all locally stored data; you'll not be able to restore a backup created with iPadOS 18.

With your external display connected to your iPad, check your settings (these screenshots for iPadOS 17 - iPadOS 18 is similar):

Settings > Display & Brightness > [Displays] Arrangement