How can I manually update Craft? If one wished to manually update for any reason, would one need to change anything outside of the the app folder, or be concerned with anything else?
Only the /app folder matters.

No other files are relevant to the Craft core operations. Make sure you replace the folder entirely, don't let the new files get merged with the old files.

Nothing else will be changed. If the update would trigger any database changes, those will be handled automatically by migrations (located in the app/migrations folder).

For more information, see the documentation on Manually Updating.