Can I leave the Story task in the backlog and add the story's subtasks to the current sprint board? Not just tag with the correct fix version but to actually see it on the active sprint board?
No, there's no way to do this.
Sub-tasks are not independent items, they aren't even items you can commit to in a sprint. They only become part of a sprint by virtue of being a part of an issue that does go into it.
If you want to handle them separately to thing of which they are a part, you can convert them to issues (stories) in their own right, then they will be sprint-enabled items.
Epics are a separate thing again - they are a layer above issues (stories). They are not expected to go into sprints, they're expected to act as a grouping of many issues (that do go through sprints), ending when all their component stories are finished.
So, yes, you can (and should) handle Epics separately, and outside sprints.
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