DBError and Network Error when saving phone numbers to contacts
Your error log says:


  ** You do not have the SUPER privilege and binary logging is enabled (you might want to use the less safe log_bin_trust_function_creators variable)]


If you turn off logging at Administer > System settings > Misc (Undelete, PDFs, Limits, Logging, Captcha, etc.)  can you save phone numbers?

You can find information on the MySQL privileges needed for CiviCRM at http://wiki.civicrm.org/confluence/display/CRMDOC/CiviCRM+MySQL+Permission+Requirements 

Not all hosting companies provide these privileges without a specific request to do so (and even then some will refuse to do so).