How do I find my iPad apps that run on macOS using Catalyst? How do I find my previously purchased iPad apps that run on macOS using Catalyst? What happens to apps that I paid for separately already on macOS?
You need to either contact the developer of that app to see if they've created a version using Catalyst or search the Mac App Store for a specific app name.  There's no automated way to find out if an iPad app was updated to run on the Mac.

Right now, Apple requires a separate purchase be made.  For example, if you have a paid iPad app that the developer updated for the Mac, you would need to pay for the Mac version in the MAS, and vica-versa.

This could change in the future but a new purchase is required right now (not applicable for free apps, of course).