OK to add Mac Mini?

No Macs made after 2019 or running any OS higher than macOS 10.15 "Catalina" can serve as a video source for the few iMacs (Late 2009—Mid 2014) that supported Target Display Mode (TDM).

It is all here:

Use your iMac as a display with target display mode - Apple Support

TDM is dead tech.

iMac 2011 with OS10.13.6 is top OS the iMac can run.

What you can do to help keep to in service longer...

Mostly you just need a more modern Browser than Safari on Mojave, or High Sierra like Firefox or Brave.

https://brave.com/

Opera…

https://www.opera.com/download/requirements

Opera system requirements | Opera

Thanks to dialabrain…

Firefox 112 is now the current version. runs in High Sierra.

https://www.mozilla.org/en-US/firefox/new/

No, you can not add a Mac mini and use the iMac as a display.

If you decide on a new Mac mini, then you will need to have and use an external display.

No.