Question: What to do when a sprint is finished early?

At the moment our Scrum team works off stories from the backlog, if the sprint is finished early.

What happens with stories taken from the backlog? Will the stories be added to the current Sprint ? If yes, what if these stories won't be finished in time. Is the Sprint failed then?

Answer: 
Working on stretch or future sprint backlog items seems to be the common thing to do, which makes a lot of sense if your sprint backlog items are small enough and clearly defined. However, backlog items that may place the "done" code into a "no longer done" state should be avoided.

If the sprint is truly finished, tag it, prepare it for delivery, deliver it, and put your source code repositories into the "next sprint" state so there's no risk that late sprint changes will put delivery at risk.
I'd encourage slack to be used for personal improvement. Sure, pull in stories from the backlog, but make sure you spend some time on yourselves: learn a new language, practice your craft with a kata, refactor some stuff, tweak, refine or write new tools to help you out, go and talk to a stakeholder, colleague or client, find out what your QA team does, take time to understand how your UX process works. 

There is a huge list of things you can do that will provide value to your business and yourself AND improve your velocity or the amount of quality value you provide that don't involve pulling things from a backlog, try those first.