WordPress 3.9 should not be an automatic update unless someone has configured the WordPress site to do all updates automatically. Updates to major releases requires manually clicking the update button. This means if you found your site updated to 3.9 without your personal intervention, you should contact your hosts for support. Some hosts manage WordPress installations and may initiate the update from their end.
As far as troubleshooting your issues you can try this “down to the metal” manual update process.
1) First make a backup of your files and database.
2) Copy your uploads folder (in wp-content) to your root directory – this is just temporary and we’ll move it back in just a bit.
3) Delete all the wordpress files (except wp-config.php and your just copied uploads folder) off your server – yes, themes and plugins.
4) Download the latest version of WordPress.
5) Unzip it.
6) Upload this new version up to your server.
7) Move your copied uploads folder back into the wp-content folder
8) You should now be able to login to your WordPress site.
9) Carefully install and enable your custom theme. Test and confirm everything works. If not, contact the theme developer for support.
10) Carefully install and enable plugins, one at a time. Test your site after enabling a plugin. Does everything work? If not, note which plugin you last installed and enabled and contact the plugin developer for support.
At the end of this, you should have a working WordPress site and have removed or identified any broken themes or plugins.
If you’re still experiencing issues after all this, let us know and please provide details about your server setup so we can try and recreate the issue.