Cron job not working after CIVICRM update (Joomla 3.7.0 and CIVICRM
I feel like the first thing to do is try to isolate the issue (or clarify it): did civicrm work before upgrading to Joomla 3.7, it was it civicrm that you had upgraded (what was the exact upgrade that broke the cron jobs). 

It's strange because those errors look more like a session cache issue.

Did you try the "run all scheduled jobs" link from the jobs page?

Are you executing with a user that had the needed permissions?