Upgrading multiple modules at once crashes the process
If you have drush try running drush updatedb.

Errors you are getting you should restore backup of your site and update modules one by one.

You can also login phpmyadmin and try to optimize and cleanup mysql if you don't have a copy as it seems your db is crashing and it could be pending database updates. If you don't have drush try going to yoursite.com/update.php to run db updates. 

If at all possible use drush for managing your site gives you option to update as well as rollback any updates.