It feels like you already looked at the plugin site since you asked about 5.1 and the scheduler plugin there has not been upgraded to 5.1 yet. I wouls first try to see if the scheduler would work anyways with 5.1 in your test environment.
If not, I might try this kludgy workaround until it is available. Use email to send in your new issues. You can generally script a simple email and send it in.
On your time question, I am interpreting either you want to track the actual work time or just when the issue is created or closed.
The created and closed times are already there. I would use SQL to retrieve those values and do the difference in SQL.
If you want to track actual work time, I would probably build a simple REST interface to use the worklog api and keep the start and stop times local to the user that integrates better with the tasks you are trying to time. In general, people do not like to track their time to the minute detail, so supplying an interface that makes it easy is very important.
http://docs.atlassian.com/jira/REST/latest/#id122998
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