When installing an application is there any way to avoid any drag and drop?
You could copy/paste it. Might be simpler if you can't see the Apps folder right at that moment.

It will stay in the paste buffer so long as the .dmg is still mounted, unless overwritten by another copy action.

Frankly, it would be far more effort than it's worth to try automate that process, for any given .dmg with random content.