As the future sprint hasn't been started yet it can still be considered the top of the product backlog, but you are correct in saying that in the event that there are already some future sprints defined that the story will be put into the backlog of the upper most sprint on the planning screen.
I think you should be able to see this in the History (History tab in the detail view) of the issue, I don't know how you can see this in the planning view unless you use a filter which contains all the stories that used to be in the previous sprint, that might help.
If a story is not resolved at the end of a started sprint, it will automatically move to the next future sprint. If there are no future sprints, the story is moved to backlog. You can easily view if an issue was in another sprint because the Sprint customfield also holds the past sprints.
No, not in issue navigator. You can configure your board to add quick filters. After you add the quick filter, you will find a new button at the top of your board. Simply click on it to enable/disable the filter. Additional information can be found here : https://confluence.atlassian.com/display/AGILE/Configuring+Quick+Filters
The Plan tab is always available if you use a scrum board (which you are if you are using sprints). More information about how to configure your board (and the quick filters) can be found here : https://confluence.atlassian.com/display/AGILE/Configuring+a+Board and in the link provided by Silviu.
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