I would like to have Groups appear in the Assignee picker so anyone in that group is notified and can assign the issue to themselves - is this possible in the Download version if not possible in onDemand?
You can't assign to group in JIRA. You can however define a multi user picker or group picker custom field and add them in the notification schemes to act like assignee.
This will be the same in downloaded or onDemand. It won't have the same functions like "Assign to me" and there won't be a separate "Assign" operation. Also you will still have a primary assignee. But other things like notifications should work fine.
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