No, you can delete it after the installation

Once it's installed you can delete the APK.

No, at least not for the functionality of the installed app. It's just the install file, like the .exe setup on desktop.

if you're installing from github you can use obtanium and it will also keep track of when you need to update

Nope. You can delete it.

The only reason to keep it would be in case you need to reinstall it, or if a new version may break a feature you need.