AirPods Pro on Mojave
I'm have been using the AirPods Pro with Mojave, so it is definitely possible to use them together.

I didn't need to manually pair the AirPods Pro to the MacBook Pro running Mojave. It paired them with my iPhone, and it automatically synced to the MacBook Pro.

Squeezing the stem of the AirPods to enable/disable ANC, play/pause, forwards/backwards in tracks all works without problems.

There's no setting for ANC in Mojave, but I can control volume up/down like with the regular AirPods.

Switching between devices seem to work the same as with the AirPods. I.e. I can be listening to Spotify on my iPhone, and then on my Mojave Mac click the speaker icon in the status bar and select the AirPods - and I will be hearing sound from the Mac instead of the iPhone.

Mojave also supports displaying charge status of the AirPods Pro in that menu - similar to the AirPods.

As far as I know, Audio Sharing is an iOS-only feature - so that wouldn't be available with either Mojave or Catalina.