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?

If you send your iOS device’s UUID to an app’s developer, and they send you an Ad Hoc signed version of the app, then you can install that version of the app on that device via drag-and-drop using the Finder.

If you install (or update) an app on the iOS device using Apple’s Configurator 2, and keep a copy of that app before it is auto-deleted from your Mac, then you can (re)install that app on that device via Finder drag and drop.

The Finder alone has no connection to Apple’s iOS App Store (AFAIK). So, strictly, using the macOS Finder only, on Catalina, no.

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.