If I copy an application already listed under Purchased to the Applications folder, will it be able to update through the App Store? If I copy an application already have in my Apple ID (Edit: for example, Xcode) from another computer to the Applications folder, will the Mac App Store be able to update it?

Or will I have to do something manually in order to enable updates from that app?
Yes - you can side load apps to Mac and as long as it’s on the filesystem (doesn’t even have to be in /Applications or on the boot drive) - the App Store app will see you have that app installed and offer updates when they are available.

There is no installer, receipt or other accounting needed to indicate an app should be managed / updated once it’s copied on to the filesystem intact (with the intact part being key).