Where to leave finished User stories

I'm not sure that there is a "normal" way to structure the Sprint Backlog. There are multiple ways for the Developers to create and maintain their Sprint Backlog, but these are often constrained by the tools that the team is using. The tools that you use will also determine if you even have a "To Do" or "Done" column - this seems to suggest the existence of a Kanban-like board, but that's not a requirement for a Sprint Backlog.

I second @Thomas Owens' response.  You are asking for Scrum Process advice but the Scrum Framework never addresses processes and leaves that to the individuals that are using the framework to decide. 

The way you have described is something that I have seen teams use successfully in the past.  But I have seen a lot of teams use different processes just as successfully.  Talk to your team and as a group decide how you will process the information.  But as a Scrum Master you need to help them understand the need for visibility and transparency of the information while respecting and utilizing the various events, artifacts of the Scrum framework.  Do your best to use whatever tool to your benefit and not let it dictate your actions. 

is it correct that from these stories we come up with tasks to fullfil those stories and put those on the 'To Do' column right?

That might be what the tool you are using constrains you into doing. It could be better if the Developers planned a technical forecast of work for meeting the Sprint Goal. There may be work required to craft a Done increment where all stories have to be integration tested, for example.

Then the question is. Once the User story is completed. Do you leave it on the place of the sprint backlog and you just mark it as finished? or do you put it in the 'Done'Column as well.

How do you distinguish between completed, finished, and the commitment to Done?

Then the second part of the question will be. Once the sprint has finished. What do you do with all those finished user stories. Is there a proper/normal way to store those somewhere? 

Does the Definition of Done require them to be archived somewhere? Otherwise, remember a user story is a placeholder for a conversation about a potential requirement. A user story is over when the necessary conversation is over.

What feels 'right' to you? What makes your progress and plan transparent to yourselves (and perhaps others)? Would you have this impediment if you were using stickies on a wall?

Whatever you decide is the answer....is the correct answer.