Please post your website url. I need to see it so as to provide a solution for your first problem.
As for the Theme Options -> Blog -> Blog Archive/Category Sidebar, is working properly.
It is for selecting the sidebar that will show in your archive view. The view that shows up when you click one of your archive or category.
Theme Options -> Blog -> Position of previous/next posts links, this is working properly. It’s for positioning the next or previous post link when you are viewing the full article (single.php)
Theme Options-> Extra-> Position of navigation links, this is also working properly. It’s for positioning the Older/Newer Posts links on blog page.
If the Theme Options you mentioned are not working properly, I am sure there will be other users reporting it.
Please kindly clear your browser cache or any cache plugin, so that you site is updated with latest data.