Jira IS a ticketing system, and a manual task tracker, and several other things because it's an issue tracker.
I think you need to explain your requirements better.
I am new to JIRA and my task was limited to setup but my client asked me for these two features. His requirements in his words as follow
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Jira has a general rule that the "assignee" is "the person currently working on this issue" (I say general rule because you don't *have* to use it that way, but it is a good way to work)
To log time, they use, um, log time.
Ticketing system - that's not quite what a ticketing system is, but you can easily set up jira to "create or comment" via email.
In short, you don't need plugins or fuss, just to read the configuration docs. (Although, I'd be tempted to reach for the JEMH plugin if the incoming mail handlers can't do waht you think you want them to)
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