1. Controllers don't work with scroll bars. Generally when you have something selected with a controller that happens to be inside a scroll view, its UIDragScrollView script will handle controller-based scrolling automatically. Check your UICamera -- you need to have the Pan axes set (usually to the right thumbstick).
2. Did you forget to link it with your scroll view? Check your UIScrollView script. Also note that you may simply have the background of the scroll bar in front of the foreground, or just forgot to add a collider to the foreground altogether.
Scroll bar showing? Just check its alpha. If it's 0, then it's not showing.
When in doubt, look at the controls that come with NGUI. Both the background and the foreground need to be linked to the UIScrollBar as "Background" and "Foreground" fields.
Check your panel. If you turned on its static flag, then there should be a warning on it explaining to you that moving widgets underneath the panel will have no effect. That said, why are you using TweenPosition? If you have it linked as a scroll bar, when you start dragging it, the scroll bar's logic takes control of it, and your tween position won't work, as the two of them will fight for control of its position.
I don't understand what you mean by your first question. Scroll bar for each element in a list?
(http://skullforge.omc-games.com/wp-content/uploads/2015/08/Skullforge_Dev_30.jpg)
I am also not quite clear on the last part... what do you mean by redrawing the list moving the starting point every time the player moves the cursor up or down? Why do you need to do any of this? NGUI handles scrolling via UIDragScrollView, either using the mouse, touch, scroll wheel or controller.
NGUI has an option to that on the scroll view script. Hiding the scroll bar if it's not needed is one of the options.
"best way to handle moving the list up and down (manually) as the player scrolls up or down through the list" <-- this also doesn't make much sense. Why do you want to handle moving the list up and down? Scrolling will already do it for you. You don't need to do anything. That's the whole point of having a scroll view, no?
Did you set the Pan axes on the UICamera? That's what's used to scroll using the controller. Generally you'll use the left thuymbstick or D-Pad for navigation and right thumbstick to scroll/pan. Default Unity project sets Horizontal and Vertical axes to the left thumbstick, but the right one remains undefined. See this topic for more info: http://www.tasharen.com/forum/index.php?topic=13047.0
As for hiding the scroll bar, select the UIScrollView script and underneath the "Scroll Bars" section there is a "Show Condition" field. Set it to "Only if needed".