The linked page has an HTML container for the featured image, but there is no actual img tag within. IOW your theme is not outputting any image. I suspect this is because the post’s related featured image data is missing or had become corrupted. This is likely the same situation for NextGen galleries. I think the related images are managed in different ways which unfortunately points to more widespread issues with data.
Since I cannot investigate your media library it’s difficult to speculate but faulty data could be the cause here as well. A couple things to check before more strongly suspecting data corruption: Check both your PHP error log and your browser console for errors while the media library is on screen.
Can you create a new post and upload an image for it as featured image? If so, this almost certainly means data corruption within older posts.
Another test would be to create a new, empty DB for WP. Alter your wp-config.php file’s DB credentials accordingly (keeping a backup copy of what you have now). Request /wp-admin/install.php and go through the initial setup for a new site. If this new site appears to work normally, it confirms corruption of your primary DB. If this new site also exhibits similar behavior, then your re-installation attempt was faulty and you should try repeating it again. The process is essentially the same as a manual update, so you can carefully follow the manual update instructions.
