The way I do is to put the site on maintenance mode and then replace the files as you describe via FTP. No need to activate the default theme as such; putting it in maintenance mode ensures no one is accessing the site while the files are being replaced. The theme options should not get disturbed during this process.
Best way to manage this is to move all customizations (PHP/CSS at least) to a child theme so that you are free to upgrade in future. Appdev is mostly child theme friendly and hence you should be able to replace the built-in functions with your own in the child theme functions.php or should be able to override the classes in the parent theme. Requires a bit of work initially but once done, it’s a lot less headache than redoing all the customizations to the parent theme. The theme comes with a sample child theme to help with this. You can accomplish this move to child theme in the staging server, test it out to stay safe when you move the changes to the actual server.