1. The first column will list all the work items that are marked as the plan Items in the project area configuration. Hence it makes sense to rename this column as Plan Items instead of Stories. In the default scrum template the Epic and Story work item types are marked as Plan Items. The article https://jazz.net/library/article/593 has a section on Plan Item vs. Execution Item which explains further on this.
2. The Task Board is built mainly based on the Parent child links and the First column always lists the Plan Items. The subsequent columns show the child work Item based on the different state groups the child work items are in. So a Row in the task board corresponds to a Plan Item and its child work items. If you are moving the work item across the rows then it means you are changing the parent of the of the workitem. Currently in RTC the links are not captured in the work item history hence you will not see this in the history of the story.