How much cellular data does downloading an app use?

You should also check if you have auto-update on, if yes make sure also to make it wifi-download only. Furthermore VDO streaming do also take a lot of byte especially if you streams in HD, sometimes you can cope with lower resolution. Loading/Downloading pictures, VDO and files sometime cost a lot of data if the person sending/uploading doesn't make them mobile web friendly.

Merely downloading an app should not cause this.  There should be a way of looking at your data usage on your phone by category and possibly by app.  I'd suggest contacting your provider to help you analyze your usage, they likely have tools that will help.  If you are streaming music/video the amount of data used can skyrocket.

This entirely depends on the app, some apps are just a few megabytes (1024 megabyte = 1 gigabyte) and some are a few hundred, half a gigabyte or even more than a gigabyte. Especially games are quite large. However if you download apps from the App Store on your cellar data there is a size limit of 50mb, otherwise you'll need WiFi to download it. In this way it safes you from using large amounts of data on app downloads.

The size of the app is what is downloaded.  I don't know if you are Android or IOS but if IOS , turn off Wifi assist, automatic app updates etc.Never owned an Android device , I would be too tempted to root the thing.