Stage changes will not, by themselves, effect any activity in the system. Putting somebody in a stage which as the title "Do Not Contact" will not block you from communicating with that prospect, it only helps you bucket prospects overall.
The concept of stages is purely around helping you keep prospects organized, as opposed to starting/canceling communication going forward. Here is some more information on stages.
If you wish for stage changes to effect communication, you may want to consider setting up a trigger.
If you want to ensure you don't communicate with somebody ever again, you will need them to be Opted Out, either by the prospect's own action, or by your manual action. In order to manually opt somebody out, you can follow these steps.
If you want a particular prospect to be finished in any active sequences, but still allow them to be communicated with in the future, you'll want to mark them as finished.