The Custom Post Display Plugin lets you add a widget that displays the content of your desired custom post type.
Yes, you have the option to choose Posts or Pages when you choose a custom post type. You can actually just use this widget to display recent posts or pages.
Yes, if you have the Page Attributes box enabled for your custom post type (or Post/Page), when you edit that post, just enter a number in the Order box. Number your posts according to the order you want them to appear, then, in the Widget options, just choose 'Menu Order' in the 'Sort By' field and choose your Sort order.