Hey Gerard - When viewing the scheduled jobs for a given org, you should see all jobs on the given environment regardless of owner, the user that created them and the ability to make any adjustments should you like via the elipses on the right hand side. The logfile for the job would be specific to the machine which that job is running on, not so much the user. As far recreating the job so you are running, you would be best to simply create a new scheduled job with the same scenario.
We may be best setting up time to view what you are seeing together. Please email DemandToolsCOE@validity.com if this response doesn't answer your question fully. Thanks!
