I've attempted to download Garageband on my MacBook Pro AND open it on my MacBook--both fail.

"I'm running High Sierra 10.15.7"

This is contradictory - macOS 10.15.7 would be macOS Catalina, not High Sierra (High Sierra is macOS 10.13). Ar you running Catalina? On Catalina a GarageBand icon would be crossed out, if it is a very old iLife '11 version - GarageBand 6.0.5, because GarageBand 6.0.5 uses still 32-bit frameworks, that are no longer available on Catalina. You cannot reinstall this older version on Catalina. You need a GarageBAnd 10 version in your recent purchases at the App Store to be able to reinstall GarageBand on macOS 10.15.7. Does the crossed out icon show a single guitar or guitar with a stomp box? If it is just a single guitar without a stomp box in the background, it is not a GarageBand 10 version, but iLife '11 and you cannot use it or upgrade it on Catalina.

If you never upgraded to the newer GarageBand 10, you can currently not download an AppStore version of GarageBand - you need to have previously purchased GarageBand 10 from the App Store and see it in your past purchases.

Which version of GarageBand do you have installed on you MacBook Pro? And which system version is it?

When you are viewing your previous Purchases at the App Store , you need to find this GarageBand version 10 in the list:

And not this older version:

You can only use this version on system versions before macOS 10.15 Catalina.

Check out your purchases in the app store, your older version of Garageband should be there.

For the midi issue:

If GarageBand isn't working on your Mac - Apple Support