The first half can be configured using an escalation rule. Set it as an immediate escalation, then in the Issue Criteria tab, select "Open" from the Statuses section. Under the Actions tab, below the Workspace/Address Book Fields section, check the "Edit Workspace or Address Book Fields" box. Then click the Configure Field Edits button and set the Process field to "Just starting." Save it and that should take care of that part of it.
But I'm not sure what to do with the date/time stamp you're looking for and maybe someone else knows. I was hoping that if the Completed custom field were a Date/Time field, you could use the same process as above except using the Complete status and it would allow you to to check the "Current Date/Time" box (like it shows when you open a new issue). But I tested it and that box doesn't show and only allows you to pick a specific date/time, so I'm not sure how to accomplish that. Hopefully this helps a bit. Thanks!