On a Mac just double-click the ZIP file to extract the contents.
To be safe, I would download copies of your themes and plugins folders on the server. Then remove all the themes (except say twentythirteen - a default theme) and plugins for now.
Once you have your site up and running, carefully enable your custom theme and test to see if it works. If so, great! If not, contact the theme developer for support.
Then once you have the theme working, slowly install and re-enable plugins one at a time. Test your site after you enable a plugin. If everything works, great! If you run into an issue with a plugin, contact the plugin developer for support and let them know about your WordPress setup so they can be able to recreate your issue.
At the end of this, you'll have your WordPress site back and you'll have identified any problematic themes or plugins. And you'll know a bit more how WordPress works and how to fix issues! Congratulations!
And yes, I understand its frustrating to have your site "broken" after what seems like it should have been a simple update! One way that you can better handle these types of issues is to setup a local testing server on your Mac using MAMP to run a pre-configured Apache, MySQL, PHP server. Set up a WordPress site on it and configure it to match your hosted site. Then when updates come out, you can test it locally first before pushing it to your live site.