Is it possible to download an app and install it on iPhone/iPad by using Finder Actually, after updating to Catalina iTunes is divided into Music macOS application and Finder abilities. So for backup and etc, we should use Finder, now I want to know:

Is it possible to download an app from App Store and install it on iPhone/iPad by using Finder?
No.

Firstly, you can't access the iOS App Store on MacOS, so you can't download iOS apps onto your Mac. You can of course download apps from the Mac App Store, but they won't work on iOS.

Secondly, there's no general mechanism to 'side-load' apps onto iOS (well, there might be, but not without jail-breaking your iPad). 

There may be methods that require the co-operation of the developer, or Enterprise device management tools.