I haven't used Patch a lot, but I think you need to repackage that .DMG into a .PKG and deploy that instead. This sounds like the same thing we had happen when we tried to push MS Teams; the DMG just had the .app and an alias to /Applications, but when you install that .DMG it just dropped the .app in the root folder. This may be a job you can do with AutoPKG; I have heard about it but not used it yet. I think Patch Management will handle shutting down Firefox.app for you, but you have to get the file placed properly.
IMO, Jamf still has a way to go in order to make Patch Management truly Enterprise-ready. Other platforms for Systems Management I've used on Windows (Symantec Altiris, in particular) would automatically download all the files, distribute them to your distribution servers and create the command-line for the install (this was on Windows, but same concept here).
That is expected behavior with most DMGs that present you with a drag and drop install of an app.
Open the DMG, complete the drag and drop to the Applications folder. Launch Jamf Composer and drag Firefox from the Applications folder to the left column in Composer. Build as a pkg and upload that to Jamf Admin or the cloud if not hosting on premises.
Or, you can deploy a script to install Firefox. Search Jamf Nation and you will find lots of support for that method.