Will there be any unexpected side effects when I upgrade the Zen theme?
Themes don't create their own database tables. 

The changes about the theme status (enabled, default, theme settings) are stored in the system and variable tables, along with many other system settings, it's nothing you should be worried about. 

Some themes are accompanied by a special module built especially for them, but Zen isn't one of those. 

Yes, you can disable and delete old Zen and its subthemes without consequences. Set Bartik as default theme before removing old Zen out of your system. 

It's good practice to backup files before deleting, it's up to you, at least check if there are any files you might want to keep, maybe logo images.  

Then download and enable the latest Zen. Create the latest Zen subtheme using the latest Zen procedure. 

The latest Zen theme structure and procedures are significantly different from the older versions. You'll have to install some bundles and other dependencies and learn how to build Sass through Gulp (if you want Sass)... 

You might experience problems if you want to keep the old subthemes and make them work with new the Zen, but if you want to start fresh there aren't bad consequences if you remove old Zen and its subthemes and then install new Zen. 

It's good to remember to clear (theme) cache when changing things like these, even though theme registry is rebuilt on visiting the Appearance page.