To delete a single segment, you should be able to click on it and hit your delete key, or click the garbage can icon. To delete multiple segments, the delete key won't work at all...you have to click the garbage can.
You have to have permissions on the segment you're trying to delete and also the adjacent connected segments. If it's a permissions issue, please post an unlock request
The stairway is an undrivable segment, but if it's connected to a real road, it can be routed. Common practice is to disconnect that sort of thing from the real roads, but the correct thing to do is get rid of it completely, as it serves no purpose being on the map.